Class 10th Mathematics AP Board Solution
Exercise 11.1- In right angle triangle ABC, 8 cm, 15 cm and 17 cm are the lengths of AB, BC…
- The sides of a right angle triangle PQR are PQ = 7 cm, QR = 25 cm and ∠Q = 90°…
- In a right angle triangle ABC with right angle at B, in which a = 24 units, b =…
- If cosa = 12/13 then find sin A and tan A
- If 3 tanA = 4, then find sin A and cos A.
- If ∠A and ∠X are acute angles such that cos A = cos X then show that ∠A = ∠X.…
- Given cottheta = 7/8 then evaluate A. (1+sintegrate heta) (1-sintegrate…
- Given cottheta = 7/8 then evaluate B. (1+sintegrate heta)/costheta…
- In a right angle triangle ABC, right angle is at B, if tana = root 3 then find…
- In a right angle triangle ABC, right angle is at B, if tana = root 3 then find…
Exercise 11.2- A. sin 45° + cos 45° Evaluate the following.
- B. cos45^circle /sec30^circle + cosec60^circle Evaluate the following.…
- C. sin30^circle +tan45^circle -cosec60^circle /cot45^circle +cos60^circle…
- D. 2tan^2 45° + cos^2 30° - sin^2 60° Evaluate the following.
- E. sin^260^circle - tan^260^circle /sin^230^circle + cos^230^circle Evaluate…
- A) 2tan30^circle /1+tan^245^circle A. sin 60° B. cos 60° C. tan 30° D. sin 30°…
- B) 1-tan^245^circle /1+tan^245^circle A. tan 90° B. 1 C. sin 45° D. 0 Choose…
- C) 2tan30^circle /1-tan^230^circle A. cos 60° B. sin 60° C. tan 60° D. sin 30°…
- Evaluate sin 60° cos 30° + sin 30°cos 60°. What is the value of sin(60° + 30°).…
- Is it right to say cos(60° + 30°) = cos 60° cos 30° - sin 60° sin 30°.…
- In right angle triangle ΔPQR, right angle is at Q and PQ = 6cms ∠RPQ = 60°.…
- In right angle is at Y, YZ = x and XZ = 2x then determine ∠YXZ and ∠YZX.…
- Is it right to say that sin(A + B) = Sin A + Sin B? Justify your answer.…
Exercise 11.3- A) tan36^circle /cot54^circle Evaluate
- B) cos 12° - sin 78° Evaluate
- C) cosec 31° - sec 59° Evaluate
- D) sin 15° sec 75° Evaluate
- E. tan 26° tan 64° Evaluate
- A. tan 48° tan16° tan 42° tan 74° = 1 Show that
- B. cos36°cos54° - sin36°sin54° Show that
- If tan2A = cot(A - 18°) where 2A is an acute angle. Find the value of A.…
- If tan A = cot B where A and B are acute angles, prove that A + B = 90°…
- If A, B and C are interior angles of a triangle ABC, then show that tan (a+b/2)…
- Express sin 75° + cos 65° in terms of trigonometric ratios of angles between 0o…
Exercise 11.4- A. (1 + tanθ + secθ)(1 + cotθ - cosecθ) Evaluate the following :
- B. (sinθ + cosθ)^2 + (sinθ - cosθ)^2 Evaluate the following :
- C. (sec^2 θ - 1) (cosec^2 θ - 1) Evaluate the following :
- Show that (cosectheta +cottheta)^2 = 1-costheta /1+costheta
- Show that root 1+sina/1-sina = seca+tana
- Show that 1-tan^2a/cot^2a-1 = tan^2a
- Show that 1/costheta -costheta = tantheta sintegrate heta
- Simplify secA(1 - sinA)(secA + tanA)
- Prove that (sin A + cosec A)^2 + (cos A + sec A)2 = 7 + tan^2 A + cot^2 A…
- Simplify (1 - cosθ) (1 + cosθ) (1 + cot^2 θ)
- If secθ + tanθ = p, then what is the value of secθ - tanθ ?
- If cosesθ + cotθ = k then prove that costheta = k^2 - 1/k^2 + 1
- In right angle triangle ABC, 8 cm, 15 cm and 17 cm are the lengths of AB, BC…
- The sides of a right angle triangle PQR are PQ = 7 cm, QR = 25 cm and ∠Q = 90°…
- In a right angle triangle ABC with right angle at B, in which a = 24 units, b =…
- If cosa = 12/13 then find sin A and tan A
- If 3 tanA = 4, then find sin A and cos A.
- If ∠A and ∠X are acute angles such that cos A = cos X then show that ∠A = ∠X.…
- Given cottheta = 7/8 then evaluate A. (1+sintegrate heta) (1-sintegrate…
- Given cottheta = 7/8 then evaluate B. (1+sintegrate heta)/costheta…
- In a right angle triangle ABC, right angle is at B, if tana = root 3 then find…
- In a right angle triangle ABC, right angle is at B, if tana = root 3 then find…
- A. sin 45° + cos 45° Evaluate the following.
- B. cos45^circle /sec30^circle + cosec60^circle Evaluate the following.…
- C. sin30^circle +tan45^circle -cosec60^circle /cot45^circle +cos60^circle…
- D. 2tan^2 45° + cos^2 30° - sin^2 60° Evaluate the following.
- E. sin^260^circle - tan^260^circle /sin^230^circle + cos^230^circle Evaluate…
- A) 2tan30^circle /1+tan^245^circle A. sin 60° B. cos 60° C. tan 30° D. sin 30°…
- B) 1-tan^245^circle /1+tan^245^circle A. tan 90° B. 1 C. sin 45° D. 0 Choose…
- C) 2tan30^circle /1-tan^230^circle A. cos 60° B. sin 60° C. tan 60° D. sin 30°…
- Evaluate sin 60° cos 30° + sin 30°cos 60°. What is the value of sin(60° + 30°).…
- Is it right to say cos(60° + 30°) = cos 60° cos 30° - sin 60° sin 30°.…
- In right angle triangle ΔPQR, right angle is at Q and PQ = 6cms ∠RPQ = 60°.…
- In right angle is at Y, YZ = x and XZ = 2x then determine ∠YXZ and ∠YZX.…
- Is it right to say that sin(A + B) = Sin A + Sin B? Justify your answer.…
- A) tan36^circle /cot54^circle Evaluate
- B) cos 12° - sin 78° Evaluate
- C) cosec 31° - sec 59° Evaluate
- D) sin 15° sec 75° Evaluate
- E. tan 26° tan 64° Evaluate
- A. tan 48° tan16° tan 42° tan 74° = 1 Show that
- B. cos36°cos54° - sin36°sin54° Show that
- If tan2A = cot(A - 18°) where 2A is an acute angle. Find the value of A.…
- If tan A = cot B where A and B are acute angles, prove that A + B = 90°…
- If A, B and C are interior angles of a triangle ABC, then show that tan (a+b/2)…
- Express sin 75° + cos 65° in terms of trigonometric ratios of angles between 0o…
- A. (1 + tanθ + secθ)(1 + cotθ - cosecθ) Evaluate the following :
- B. (sinθ + cosθ)^2 + (sinθ - cosθ)^2 Evaluate the following :
- C. (sec^2 θ - 1) (cosec^2 θ - 1) Evaluate the following :
- Show that (cosectheta +cottheta)^2 = 1-costheta /1+costheta
- Show that root 1+sina/1-sina = seca+tana
- Show that 1-tan^2a/cot^2a-1 = tan^2a
- Show that 1/costheta -costheta = tantheta sintegrate heta
- Simplify secA(1 - sinA)(secA + tanA)
- Prove that (sin A + cosec A)^2 + (cos A + sec A)2 = 7 + tan^2 A + cot^2 A…
- Simplify (1 - cosθ) (1 + cosθ) (1 + cot^2 θ)
- If secθ + tanθ = p, then what is the value of secθ - tanθ ?
- If cosesθ + cotθ = k then prove that costheta = k^2 - 1/k^2 + 1
Exercise 11.1
Question 1.In right angle triangle ABC, 8 cm, 15 cm and 17 cm are the lengths of AB, BC and CA respectively. Then, find out sin A, cos A and tan A.
Answer:We have
![](data:image/png;base64,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)
Given: ∠ABC = 90°, AB = 8 cm, BC = 15 cm and CA = 17 cm
We know sin A is given by,
![](data:image/png;base64,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)
⇒
[∵, perpendicular is the side opposite to the angle A & hypotenuse is the side opposite to the right angle of that triangle]
⇒
…(i)
Also, cos A is given by
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKUAAAAtCAMAAAD4FltdAAAAAXNSR0IArs4c6QAAAMZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZjqQZmZmZmaQZma2ZpCQZpC2ZpDbZrbbZrb/kDoAkDo6kDpmkGYAkGY6kGZmkGaQkJA6kJBmkLaQkLbbkNv/tmYAtmY6tmZmtpA6trZmttvbttv/tv/btv//25A625Bm27Zm27aQ2////7Zm/7aQ/9uQ/9u2/9vb//+2///bn5F03gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADa0lEQVRYR+1Y21LbMBSUnAYwLZAmpaUXSEyhN+wATXGMXbnW//9U96ykhMy0M3nAjTNFD7YiK/Z6dc7RepV6aoGBTPfutoCNbG8LQNpkuAUom1fpFqCsej+mus/QLGLd/3ZYKlUfa/3sDCMGnZddeIl899OsGUls5tFYVRq9ZjQo6wSdAiN5J5Iri1Jlk51SmRgBauKJP+RD96MTKJsRsBFlJqSZfUQpqMWyK5XtlL8Kwb7xFmABKVed62viPsDaROvo4PvGIQZYlZ6AQJLqimd9GUa6gJGr6g5EmWuJzYGDyxE170ChsklvZm+QQIKrLD6MhuZ91UuZSjbZLe0NIG+8mZNbFMmZ4Kji6ExlelD+fI14lGqJshmdMo+eWlcYYKHrfMv1FqBs3nzpAkrRJJHsrDgfSibW5xgYhFWeTrqwy5p4rOrRBNUL5wSbl8ShvQj8mcOyCyi5MaBR9ItC4XYAcBy15ynU1srWkOnQVsfbTDC3RYlGkTOPU73c/iv8rtZAs0DeSsejCyidEiheeGGNq/LUNVC2SeSCwxUuMXofO90CmQouoV0etE2seNBRrnZLXDIkXbSa504JrqBsmbY/3z7XY2WvUzA2RI5T9SPZj4VL/wZzkVqbbnMIliuJxlhDrEi5ROcIHXvJOlqtF5hP/sXjLeS6/sW68x4P2cM7retfFKsVox0wf70r/ItLltl5jCO+DffsFPZAEUtsswaLNpdvRi2ugbc0MGen5Bz2/BbSnu8h/oV8yZgzVi7sYvO06n0du2/wPLpSjkW/I3tLw82RMig9THQfca35HuJfNFAtvoCx1PJzUWQAPQ16ML66LS0NzqGIEFXDy+5iG74HIfgdX/gQAeB4E5Tkk1O8blhaGnLJZZRA5eX2fA/6F17j4WQ/ApZ/OH5RHpBQ9yJLS4NzQox4d6Y938MFldv90TdCjn948AYI0L1IUGJ+Dv/M9yTp7fkeC+tC0O2n1/IwwpLD0s7yybO0NDiHkSFQTSxb9KAt34NriuS5PyVTb1l9RCLc008DDDvlWoOs+gQhECwNcpvt2c93iN/63QV6t635Hp4tCoGFX5UdwbgYC94G31QHdDeMszmCpeFdOBnDnEGZo/ePfA+mDqllmHa0zR06pkNH2zSlwA/J002UNnGRKIniovS/ab8BLqyGB4sNSJEAAAAASUVORK5CYII=)
⇒ ![](data:image/png;base64,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)
⇒
…(ii)
Now, tan A can be found out by two ways:
Method 1: tan A is given by,
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Method 2: tan A can also be written as,
![](data:image/png;base64,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)
⇒
[from equations (i) & (ii)]
⇒ ![](data:image/png;base64,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)
Thus,
,
and
.
Question 2.The sides of a right angle triangle PQR are PQ = 7 cm, QR = 25 cm and ∠Q = 90° respectively. Then find, tan P – tan R.
Answer:We have
![](data:image/png;base64,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)
We don’t need to find hypotenuse (PR) in the ∆PQR as tan θ = perpendicular/base.
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
And ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
tan P – tan R = ![](data:image/png;base64,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)
⇒ tan P – tan R = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAC4AAAAfCAMAAACvQQ8NAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADo6ADqQAGaQAGa2OgAAOgA6OjoAOjo6OmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZpDbZrbbZrb/kDoAkDo6kDqQkGaQkLbbkNv/tmYAtmY6tv//25A625Bm27Zm29uQ29u229vb2////7Zm/9uQ/9u2//+2///b0PHgFAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABMElEQVQ4T91UyVbDMBAbt0ADtIE2hcQFuoRQss3//x6Ss7xsh3LopXNzLGuksRWRf1eyXaUi58AYH2djY8xs3yXJvQjLPDCzdxE9vgAs5eZLEsJi7nVLd0/4VK5DyT1sL9q9/HEKHocW8Gx+EjmI2tX2mWKo47UWQ1FNZb4STqrCooH/e7auQ9KgILZcwwIqKt9OasPPlJ6WATRTwD3o45ZTLdtURefGcJ/+IMZPlewZ0FkkutujY8+uE4MqArAUHGAK5643BoU5hr3R/Hi0qfau/3kwwJtbVrdxad2c/WsZ0iNfvloOdjER237f72Xg4B+pe8ij2I5Utsnl878cTnInphvb8Qga9qz+PwyCNTxQwx05qxvbMbkeHlyIHflEbPsnXOyR34p8IrbXusCK9w8fXhhogUCLrwAAAABJRU5ErkJggg==)
⇒ tan P – tan R = 576/175
Thus, tan P – tan R = 576/175
Question 3.In a right angle triangle ABC with right angle at B, in which a = 24 units, b = 25 units and ∠BAC = θ. Then, find cosθ and tanθ.
Answer:We have
![](data:image/png;base64,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)
In ∆ABC, ∠ABC = 90° and ∠BAC = θ.
Using this information, we can say
AC = hypotenuse of the triangle
BC = perpendicular (side opposite to the angle θ or ∠BAC)
Using Pythagoras theorem,
(hypotenuse)2 = (perpendicular)2 + (base)2
⇒ (25)2 = (24)2 + (base)2
⇒ (AB)2 = 625 – 576
⇒ (AB)2 = 49
⇒ AB = √49 = 7 units
So, we have AB = 7 units, BC = 24 units and AC = 25 units.
Thus, ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
And ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Thus,
and
.
Question 4.
Answer:Given that,
![](data:image/png;base64,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)
But ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ base = 12 and hypotenuse = 13
So, using Pythagoras theorem, we can say
(hypotenuse)2 = (perpendicular)2 + (base)2
⇒ (perpendicular)2 = (hypotenuse)2 – (base)2
⇒ (perpendicular)2 = (13)2 – (12)2
⇒ (perpendicular)2 = 169 – 144 = 25
⇒ perpendicular = √25 = 5
Using perpendicular = 5, base = 12 and hypotenuse = 13, we can find out sin A and tan A.
Sin A is given by
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
And, tan A is given by
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Thus,
and
.
Question 5.If 3 tanA = 4, then find sin A and cos A.
Answer:Given that, 3 tan A = 4
⇒ ![](data:image/png;base64,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)
But ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ perpendicular = 4 and base = 3
So, using Pythagoras theorem, we can say
(hypotenuse)2 = (perpendicular)2 + (base)2
⇒ (hypotenuse)2 = (4)2 + (3)2
⇒ (hypotenuse)2 = 16 + 9 = 25
⇒ hypotenuse = √25 = 5
Using perpendicular = 4, base = 3 and hypotenuse = 5, we can find out sin A and cos A.
Sin A is given by
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEoAAAAeCAMAAAC8EXiiAAAAAXNSR0IArs4c6QAAAH5QTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgBmOjoAOjpmOjqQOmaQOma2OpDbZgAAZjoAZjpmZrbbZrb/kDoAkDo6kGYAkGY6kGaQkJA6kNv/tmYAtmY6ttvbtv//25A625Bm27Zm27aQ29uQ2////7Zm/9uQ/9u2//+2///bWyWMXgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABQklEQVRIS82Ub1ODMAzGG3TWbeIcK/7ZcJ0KhXz/L2gSUeCu0HrXF8uLwpXyS/L0uSh1beH0IVFJWK5ToWxhEqGaHBOhuv0ZTfGaQiwLHKsUKGJMG2zAqxyaVR3O96VvzsOpGZSFGFQ4mVLd/i0e1ZYAWV43QA3yctGQD0mqgx2XvpidtcBnyized3pzVHZQzW3reFS3eyTC9g8lPNqSwJLA2XFUy0kunmOy/XOigs07P/uqCCV0iYZeGs8/0z57Nm1+rOGWbtKD6nYz6ecl+9R3XpTl6qYmWWqQZVKnQfZRg+6ebRc/ipwuVPtEVUl2WXrZ0dCuUhf4VS7kw7bUAA98geQpWVgh+htfIBO4765C1Gv4zlMmaJm4Qm2iaUzZUqKowdEIiGtl9lS8+4KJ0MS6bxHF8yNRVViRF4pg4f848A0IjhafgMhKhwAAAABJRU5ErkJggg==)
And, cos A is given by
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Thus,
and
.
Question 6.If ∠A and ∠X are acute angles such that cos A = cos X then show that ∠A = ∠X.
Answer:We have
![](data:image/png;base64,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)
In this ∆AOX,
cos A = cos X
and cos A is given by,
[∵,
]
Similarly, ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ AO = OX [clearly, since denominator from either sides cancel each other]
Now, if sides AO and OX of ∆AOX are equal.
Then, ∠A = ∠X [∵, In a triangle, angles opposite to the equal sides are also equal]
Hence, ∠A = ∠X
Question 7.Given
then evaluate
A. ![](data:image/png;base64,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)
Answer:We have been given that,
![](data:image/png;base64,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)
And we have to solve for
.
We know the formula: (x + y)(x – y) = x2 – y2
Using this,
![](data:image/png;base64,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)
Also, we know the relationship between cos θ and sin θ which is given by
cos2 θ + sin2 θ = 1
⇒ cos2 θ = 1 – sin2 θ
So,
⇒
As we know,
![](data:image/png;base64,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)
So, B = 7 and P = 8
By Pythagoras theorem,
H2 = P2 + B2
H2 = 82 + 72
= 64 + 49
=113
H = √113
As we know,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAASYAAAAzCAYAAAA6qKVGAAAN10lEQVR4Xu2dBaz8uBGHvyszM1+ZmZmZub2qzMwqSYVrqzIzMzMzMzOzVGZm0KezJTdKdrO7cZLdnZGe/nfvJbYznoxnfgM5gKDgQHAgODAzDhwws/XEcoIDwYHgAKGYphGCwwO3AY65ZPrvAO8FfjbNMmPWPeDAYYBbA8dd8qy/AT4MfBH4b22+hGKqzeH28Y8E3B44OnAL4ITAM4GfFpcfFjgPcAngKcB9gb9Ps9yYdYc54CF5W+BYwM2A4yVZVBFlUk8cCFwb+BRwE+BHNXkSiqkmd5ePfRTg88C/gHMBf2m55V7Ao4BHJOW0fNS4IjiwOgeOkJTOP4ALAP9sGeI6wKuA9wBXSHK7+kw97gjF1INJFS85C/Al4DWAm95GJwN+COjWnQP4U8X1xND7y4HTJlnUcr9LBxuOAXw5WVenAX5Si12hmGpxtt+4twSendy553Xcckrgu8CPARXZb/sNHVcFB1biwA2AlwFXBt7acecJgG8ky/5MNWUxFNNKezf4xa8FrgmcNZ1EbRNcBXhTMp8vB/xn8FXEgMEBeC5wLeAMDayz5I0Bm2eMASuEYppOJAXAPwscCjgn8OeWpbg/LweuBlwe+OB0y42Zd5gDhwM+DfweuHjH4XdB4I3AG4A71g7EhGKaTto0hb8CvC6dVG0r0ax+PnDnpKCmW23MvMscOBXwVeCRwAOLB/XQFEowane2FB3Weo90gR2WhptziPnspgs4Zjo0ILB4K0Dg+2HAF3aYD/Fo03PgesArgGe1ANqmE5hKYPT4hcDfxlhuWExjcLl9Dl2066dT6FeNS74FfCgB3tOtMGbeFw48B7gGcHrgly0PbSLw+5OLJ6Twi9qMCcVUm8Pt44svfQYwd0QT+Y/TLCNmDQ5gIu8nUxpKF74km8wO17J/NHDv2nzbVsWkeSmjTFD0xxDmi2sza8DxPZnMB3k3cMUBx92WoQRbTSg9OXBiwOjkD7Zl8Y11brssiiEpi0bb7rFgD64OvD4FYFRgVWkdxeQprz96OuDSE7gbWhvm/LwEeAsgJuO/90l1PFUZNtDgN07++j2Bxw405jYNc6KEoV0EuHBKl9B93TbaBVk0RcAEXyO/AttdlDFRo3K6fVVpHcV0fEAhOhpwIeBjVVf4/4MbJbBu7N8pUpWjA0au9I2rm5gDPatK9SDAEOzHBxpzG4d5HHCpVBNoKcQ20a7I4tMBD8ozpgqDtj3w8H8ncDFAy6krAXOw/VtHMTm5CzQL9NUjJ/yZjKjJaWmGmdCZjG6dIgn5YMypNJAWp/iSkY6q2bOV1j/UsAq7eVnfTJnvQ4071ji7IIsZX1LJWjDeVh8nPy2XMivcKgVzmKon+a6rmMba/HIesSQrmy0gNK+nJCNcp07FhxbEzpk8mWwdYZTDTO7qOSEzZYbunNigaRGvnOkau5a1K7KY8SWhGbtcNEmFpTX1mJRKYI5Tl/IadAu3STHZkkFs6dwpYzozQubZJ0ZA1aroOSome96Yzm9LCduYiK1Y//bS1D5CV7T6KTSo5Gw+2JVSpbr1f9/bfLhRR9hmWdRS9TCw1Y5QgjjxB9JPyURBfb0QIRItJZOBR6OmYvIksLDUU9xK4pyvIECpCWdYW8VgJOXYKT09XyMQeHbgJMBxEjjt9eJQ/tjf5c1rNj3zxVb5HDWZnH8tOOQ6zFo1snCZ0Ti3/RO59+6lEZavA29boBxVpu6tlk2uKNfiuWraE5vZfW4BS9rmenxS0Odb8xR2TEt5XL+H0ycS3in+2CSvNQooBCHpQloO1LRWl8m/BdQhiyPIfqmY3FyxGvv+6PdnOmkKE+p22C/oDkkgPfXL4lOtAf92WeD8KWp3yaSJHU/lJuhrFfOqgLmFhWaeviBlSpes0UpyvKel+Udg29ZP4eGhYlChaIV60KjU/bENy8HFE7qfWqE2sTNcrCLwR1kQEPVkNSpquoYna5O65vIAVDncbg1ummZgTs070r8C5+aDmbBqlPPnxZi6+GbP6664TmXeUh+zne+f2sl4eR/5/3Uqcg1ZXGPTVrmlVEyabeaTaN41OyU+CHgy4MZImn/m4LRVxee/GQpX0f2uWFBG8zXjVyGFV8WziBTKRViF2M5HkyW4ytzltVpqWhlfW3eAGdynolAh2XzO/ckk/8Tq7E74ovRLX1bxvCemYIfRWGVErMEeUZn8fzOCPXxKi2XZXIadDT+vQkaFzYr3kCsVqFa54W5B2julAZXp96XOjM0CaA9NXWitKHOo+sp/yOIqu7XmtaViUoh0iQzhCnaV1e5aQAK22YW6aDKH2xRT/pvCYWi/JF8EI2qrmu92zVOIdSdcY0kKo66GEa6q7T578LgsgOxx+WCXPLjnSEZhLBq2xECelUCmbpl8dk9zTpGWlC+w1pD/qtA8VHT7StKSNa+tVEzL5lIJLgpRdz2SSlJA1iTV0jJybi06D9GnJstIJaXCaSpMx9Yl09LWjVUZ95X/kMXFwtZXFheOUiom/1sz+G6pUM80daNgvvhaGqU/3kcxae43TykVk/cq/CVOtGiRgnUfSeH1ZnsQe2arMI3u2Opz6gjX3BWTVs3bExBvwWZJvvC65+WhoXsuiSM+NLlKHixlCU0OOSsrFiRnWjaXLrjW+SrBityKWEBWZbMoYCC4q9LR3dRtayMtbCEKoQI/+LBM/kMWl5+Agysmp9R0FwO6UVIgAtqSJ9BdCyHqo5g0kTW5S1pHMR0xKR9byzbLN8RE3pWarXlaBi3mgBm+4iulVeQdWbl4GLVhPjnnSIXQzPrV6hH4NoO4TLxbNpfBjK4Wrl1PId6pVWyqhRbcIjpvqgETQ2sLhXuv8mhGs8pYxbpM/uWTB2HIYuU3rbSYVAAqoowjKYya537Nw1D3dZOp7JLGVkwCsp58RgZL0h0wF0MlOEo7hsr7UXN4EztVIGJA4mQljpjzWXSRxJCalHOOVCTiMiXpOglkl21/+8wlpmWUdhUyqffbCTRfVq+ldS24bkqGB20b+Sw3TRE7ratl8i/YHrK4yo6teW2pmBQscYcnNcbyGs1+BTkrhjEVUzafjagYXclkzyJPeLNSxT6W0b6D3x48vlQC99ZFlaSVbOuLMwPfTy+ySj8D2bb3FZtqYnyO6R5YVuPhpUXhWFYELJpLeRIjMntfpVHOtWgfHV+33lSWrq6fZtQb1je1xOiZEeGuQmmxMg9f3VMjfcvkX8Xs/CGLy962Df/eVExGvtycpt+veX/kBIo75ZiKyfk82bTkjPRJmtwCsoKfVkRPjS1tuA2j3J7dMRM7BXsz+bKLI5p7Ju7jdSqZ8oASf9Jt1tIqPzGVUzVyzZ/uk9aSmKT4Ytdcgu9G0XLiafMwXMSQ/Akho7/NA8m1mwLgAaZS9dNXd0+ua7PPkB94VHk+IXVu9GDuI/8hiyOIa1MxuVH63KW5bp6KfvoD0mnqsnSdzBZti8rlv7VhTI6bq8n7gt/O5z33SxEhlZAvlspRhblNxZ+6Cq5ZELdJuihGkSTdHE/yJumuivet+wknXTVdc3kn38yWN4tZjMXolbiNvFYuxIAklYcWkbiK1lBJKgdlw5dahWUk1rQSo33rzNVH5D2UdB8tJhVgLz8hJN/EoHKenErXyLAy49pyFNLfu06fUy/A3+dPaS2T/5DFQ3ZpU1lcuNelYtK01qz29NS01SR2wzwBNc01iY3SKNgKskIh4KzfrRvgiVv+TSDUBucm8pn8ZqKeCsWTStDRv9mqsw+5TnEMW83qUnr6OcYqEZ0+89S+RoVkEqpuhs9jTo6Jf/LWQ0F3Sbphaq/ri++LoEsiP82et4h5FaVePpMvteUIJh3KQ3PM3FtfXBNrda20Qt3PHPFyzVo/HkxG9EryBX94Wo/uk3hOrgRYZ65V+C9vVEQmftpE/w+pbKKZvKvs6IpqzZkC47Nq/St/YlzZ2u4j/64vZHEYWeytmFYRirh2Mw7kDwf6ooiVtH2B1xlyexRbg5goGBQcGJoDs5TFbSriHXpDphzPhFXdI62lpnuU16WlalRJK1Xgvq0X85TPEHPvBgdmKYuhmKYRLgF7s+vL8o/mSsxk9+soRoG0mPat+8A0O7N/s85SFkMxTSOIhqkFmpuJjuVqcmuNhzS+9TXNimPWXeXALGUxFNP44pbLaARrBWS7EkPz550CXxp/j/ZlxtnKYiim8UXQvkC21jWqaGSujfLnw410Br40/h7ty4yzlcVQTOOLoPk0fRMKzRULfGn8PdqXGWcri6GYxhdBc2dyIa1dN9sof7I58KXx92efZpytLIZiGlcMTay0Ot1yCZNYuzK4TXC0Ij7wpXH3Z59mm7UshmIaVxRzxfuijwZaJmILDlt82Ceo+nfix2VBzDYTDsxaFkMxjSsl1mVZo2VDNXtWt1FuQSJAbkF15C+Nu0f7MtusZTEU07hiaG2hxae6cSZPtlH+ZLMV8tanBQUHanBg1rIYiqnGlrePmfsD+ddme9ryDj/ZrEVlmxE/7hkUHBiaA7OXxVBMQ29593jZpzcSYuP/NrJa3xq6AwNfGm9j9nCm2ctiKKa6UmkvK60fW8jk7+354QS/tOEng/y6iG03VFRaUXblPChF68SgbEti8zLbnQQFBzbhwFbJYiimTbY67g0OBAeqcCAUUxW2xqDBgeDAJhwIxbQJ9+Le4EBwoAoHQjFVYWsMGhwIDmzCgVBMm3Av7g0OBAeqcCAUUxW2xqDBgeDAJhz4H5O6a2EJTVWDAAAAAElFTkSuQmCC)
So,
![](data:image/png;base64,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)
Now,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGMAAABnCAYAAAAUqvyDAAAHyklEQVR4Xu2dZeg2RRTFf69dmCgqNnajgq0oFlgottjdiCIWBnagYL1gN2IgInZ3KzbYhWAXdnJgFoZ1n93Z2dn97+wz++X5sHdmZ8/ZqTvn3mcS6eoNApN605LUEBIZPfoIhkjGQsDFwJbAXwVYrw5sB3xl7s0MXAm8M9G8DI2MKYEbgUUAgZ4nY1fgMGATi4z5gDuA3YHXJ5KQoZGxBXAR8EUBGSsBjwD7ATflQD8G2AxYD/hjoggZEhnzAJsDGwMLFJBxA7ATsETBkLQRcB+wBvBMIqMZAlMAR5q54toCMqYFXgSWNfc+zT1ueeBV4GTgpGZN8S89lJ6hyVpD07PAbQVkTA+8Biw6gozlzP27gU394WxWcghkzAnsAFxooCgiQz1H88U6I8hY2fScF8xQVbQKa4a0Q+nYyVD7jwAuA34oIUO3DgfOA1YFns9hszNwPfAmIGJ+d8AuuEnsZKwP/Gu++gycop6he7MATwO35OaFqYFDgXMTGf7f12yA9g0XGEKqyND9BYFLgOuAm4GZgK3NfPGSmXPWAv72b5Z/yVh7htqtzZtWTt/mXn9Uz8jMNH9oqNL88QlwO7Ak8IrpNdqdT8gVKxlaip5mJt08cNubIely4B/gauDjCnS1Wcx24ddMCBMwOEfhVNamrcgdMgrns4E9gBWAzxMZYRAoI2MZ4EHgCuB463EzAFrSang7IUwz/GqJdZga9bbTmGWr9gmr5RyFGwDa1KkX2GQcaDy8WwG/+MEYptRQyJgLOACY23hfhY7c4l8Ck82vduFydagHyDUiF8m+wFLAUcCPYSD1r2UoZLgioJWUesyaphfcD7zrWrhtu3Ejo208G9WfyGgEX9jCfSXjxLCvGbw2udqDX4kMP0jHigw/iCIv1deeETmsfs1PZPjh1kqpREYrsPpVmsjww62VUrGSIQ2UjlDlT/quBJkqO+3Idbg0K6CzdLlMJPORrureVhAvqTQWMgTaXsBiwNKAQFbbJb35xno/VzsVUXk5Ce8CPrLq0FGuNFSnGulOZ5zERMZuwNfAY8BVRsVRRIaLnQCWg1Cne3cCEiRkSsJMY7UwsCLwXldsxEJGHg95XqX+y5NRx04OQ6kHdfQq3VTmtc3ORFYB1gUeT2SUIxCCDA1pOwJSF9qAa954GZjfkG0PYa3yMs49YxSwGpp08icd1Z455UkiowCBED2jCFhpqKQ4mcPEcHzfKvq5ylPPAMV07A/otFDzhPRTpwB/dklEtrzr+pkhntdWz9Be43xgduAQ4P0QjXWtI/WM/yM1I/Cc2QhKXZgm8Iqvqa2ekT32HBPvIVnP3q5fdlO7ce4ZknRqt61VU14Zsg9waddC6HElQ/oqDUVaxp4JKKbPvjIyFHCpSb2TOL9xJUNLWO0lJOeUdkrxHfYlv9RxJgBH4QKdXDGTIe2TJJu2ozAPmuaWUXaKAZSCXWI3+5LcU8E08uTKZfJZJ0xEJnyWVF/OPQXRyxmojZliLN4GfjJxF7+azZqLnXqHlIRyiyheQ0p1EXC6cYNICN2ZkzDmfUbIj1WbPTkEFZL8s4mCUuSrIqI6vWIdpjoFqauHuZChw5ynTBf2bZeGD61K3vKtYBzKuZDRBQ59VxBWYRBE1JbIqILZ7f6gyHB75YFb9aVnDBxmt9dzISNN4G5YNrZyIaPxQ1IFbggkMtxw6sRqKGQoDYXEA3KLSN0hl7j2NA84oFiV09ChijAmQyBDIcVaWu4CfGBgkTdWx6dSIX5YAlVVTsMwKDvWEjsZEp9JE7tNLp2dolg3dBChleU0dIQwnFnMZMjreo9x6CkvofKEZJdc63J/S3YzSuVRldMwHMqONcVMhsQCTxgVhzJ21rmqchrWqSuYbcxkKCRA2dXWBp6siUhVTsOa1YUxj5UMO+fg4iZUQNpY6Z10+PRwyUrKJadhGHRr1hIrGZLtS/mnuUErJqnJdeKnS7onTeC3mhWVDYne1zWnYU0om5vHSoadGvVo4KwcFNuayVvZ1iQ8yK46OQ2bo1uzhljJmM7I9nXWLYCVJtW+sjy1ysKms2xdPjkNa8LZzDxWMjRnaCWlgBmlyMsnlNf8oSAYxVlkead8cxo2Q7hG6VjJ0CsqCFL5CMvIyPLUSmwQMqdhDYjdTWMmQ8OPNE9FSYMl3dSE/pDZiY9SevjmNHRHuIZlzGTMC7wBHGz+M8N+bSWLVELhIrWgbZfIqPGxVJkeaxSD8jHZiYGlIpeSUE7EsryDZTkNq54d/H7MPUNgyD91BvCb+ZUfSq50kXCQ+eeAItBcchoGB7uqwtjJyN5Pwfr6QxIteR81q6jOFYFVYFfdHwoZVe8Zxf1ERo9o6isZfVcYBhGt5b+DRIZfzxgrMvwgirxUX3tG5LD6NT+R4YdbK6USGa3A6ldpIsMPt1ZKxUpGVe7BDKwqu5Sj0OOzcs096GqnJuhDTDkKPclwyT0oMlzs1ISUo9CDiKIiIRK5pByFPSIj5SjsERmjmpJyFNYkKcQwVfTIlKOwJhEyD0lGylHoQYBdJCQZdr0pR6EHMW2RoaakHIU1CWmTDDUl5SisQUgIMlKOwhqAl5k2JSPlKAxERIjVVMpRGJiMlKMwIKB1q0o5Cusiluz9EYj1cMn/jXtcMpHRI3ISGYmMHiHQo6akntEjMv4DGpIAht+T5GcAAAAASUVORK5CYII=)
Question 8.Given
then evaluate
B. ![](data:image/png;base64,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)
Answer:Given,
![](data:image/png;base64,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)
We know that, cosec2θ = 1 + cot2θ
![](data:image/png;base64,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)
Now, To Find:
![](data:image/png;base64,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)
Dividing both numerator and denominator by sin θ, we have
Question 9.In a right angle triangle ABC, right angle is at B, if
then find the value of
A. sin A cosC + cos A sin C
Answer:We have
![](data:image/png;base64,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)
Given that, tan A = √3/1
And tan A is given by,
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ perpendicular = √3x and base = x
Then, we can use Pythagoras theorem in ∆ABC,
(hypotenuse)2 = (perpendicular)2 + (base)2
⇒ (hypotenuse)2 = (√3x)2 + (x)2
⇒ (hypotenuse)2 = 3x2 + x2 = 4x2
⇒ hypotenuse = √(4x2) = 2x
We have, AB = √3x, BC = x and AC = 2x.
Using these values,
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFIAAAAdCAMAAAAO/MsQAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OgBmOjoAOjpmOjqQOmaQOma2OpDbZgAAZgA6ZjoAZjpmZma2ZrbbZrb/kDoAkDo6kGYAkGY6kJA6kJC2kLbbkNv/tmYAtmY6tpC2trZmttuQttu2ttvbtv//25A625Bm27Zm27aQ29u22//b2////7Zm/9uQ/9u2//+2///bVRgYCQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABYUlEQVRIS82Vf1ODMAyGG2Srczo3pgx/jalzgEpL+/0/nGkLAh4blKue/aPX66VPkjdtQ8i/HYxOCaPXDuOTz3z2epOFDpGEyM3UKQ9hInCZNQLT/fpl8vn4I04GnUrIzSTvzyj2DyLwt8OQCQxB9jutLcT6wR5ZRADeImeAiasppbCokbswOTvYxIC2Sit5h5FwilpyOt+SpFaVX+b2SLHE4uPJCqm51X2QETrwmqrHUI3WdjuPHczf1E4ZJSK1Fz0YLtiJs8bq24tZ4E52AT7K1YEUS2XSi+yS+h3ffBcyUdG27+uQxJWMJK7L00icz1StdfRWg9NbUgQYpY5GT2V5yo8gBdvHW0QU4EoVHO+knpSCSJH34Gkn48S0yutPjD8CaL44Bz7F6olko67aSef8HN+YaUAOgtQIdYXLBuQImZnPy2EDSsrv0F0DYkhk4ZEGNEoH87uEpLsBjUL+/qEv8WQei1pHUM8AAAAASUVORK5CYII=)
⇒
…(i)
Similarly, ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒
…(ii)
Also,
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFYAAAAeCAMAAACBgxnEAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZjoAZjpmZpDbZrbbZrb/kDoAkDo6kGYAkGY6kJA6kLbbkNv/tmYAtmY6tmZmtpA6tpBmttvbttv/tv//25A625Bm27Zm27aQ2////7Zm/7aQ/9uQ/9u2//+2///b5w6WLAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABoklEQVRIS81Va3OCMBBMFKUPbau2EvsA+9BIoUD+/5/r3iXo0LEKTGba+xBiTPY2e5c7If65GTUGw2oupRw8+OOqrwlW6FEm0qE32GKaAlCIZCbMGzvwYUZF+XDrRJj5QGSMZCyKS8DS8HURe8LNESg5AJrG/at55Ae2nCP2hGYUpF2TGmfNKI7FCTNKjjIMw60m1pPNWUxKGZxps6/bnur+uRdssYDaEJm+U+JfrrBwWzNcR7qVVj/IFuFSlFCbvwoIpKV5rBkW06wXbOIAkHlgHEaIKNIZaOzerGKhKXMOllBY2JrrDboMArNfHtdy8lHvyfE7P3Hc7ts7qicOroa1ZSi9koFNPS5Ip1j9Fv4jbLH1M7TVQtMVctl4Na1EsPwgIic9acuyWsX5mfNiZ9NyKcxrDE4zZALQOCUW5My53OGf7rYLZfBCioZSUraWK0xuMDFPnM8oI2dj1t3rH59wipl33PXOHxfXy5JgY/PHj7lexp3Hn7leRmXcp7le5q3fWG51Lyu8dUdC3fcyKgUm6fEojyl36GXcC9r1sj4h+AY4BiVMFB0puwAAAABJRU5ErkJggg==)
⇒ ![](data:image/png;base64,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)
⇒
…(iii)
Similarly, ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒
…(iv)
We have to solve: sin A cos C + cos A sin C.
Substituting equations (i), (ii), (iii) & (iv) in above,
sin A cos C + cos A sin C = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIEAAAAiCAMAAABP9koiAAAAAXNSR0IArs4c6QAAAJNQTFRFAAAAAAAAAAA6ADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjqQOmZmOma2OpDbZgAAZgA6ZjoAZjpmZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGY6kGaQkJCQkLbbkNv/tmYAtmY6tmZmtpBmtv//25A625Bm27Zm27aQ27a229u22/+22////7Zm/9uQ/9u2//+2///bPhjTBgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB6UlEQVRYR92WW3PCIBCFwdZqr2pvpjfT2sTWaoT//+sKBCwQwtk+ONMpL8zot4eTZRdg7J+NitsBv4tOQqkAeHun8nSSqmg4cUvFA1K+jkAcJqzA9o6x+owP1ASGT9YXM+AAE265esHkwwtbDRbIQUhWKAcME2ZF+bQxc3OKHEQk1seEXKrMN20Z7B7vMylIkVgfE5/L4YZt53rlkh/lHKRIrI8JJos5cx32MTZWekaXlOVJu339MZBQodXwS2+CmC5Yk3XQIc3hdJUzgAkdLabPpgnrMepGOgkyGf9d4R60IVlSXLfn6s7Omb2M/2rOu5uZlkuRezVZHGsLYnKZrwxVd5RBlAukZKHaSkzAIaXrmTRocqGUilnPRqg7qA5UlyK57n0uC67S8DNionRvAO6V3v63jl6P3D4gkco4JJVt8i6oyyL6ILx5Jm1BEpIuiXXQ7gKSS1ZivhDIOXCVCOrKs2AidDfmm4HqgCjn52B3Y694O//iREqhRDlcG4cl1jPO8wcgJqxDMsh8UkzRuw4T1gAZVPdotCh+12HCuiCD4WOyyj4PtDYmrAMyGEiu8mWgtDFhDZDBQLKCBjDhMgClXFf5klsV1T5y+wYmbCQZZD4pJvqWyjnAhOsFKOW+kSx52IPoz6h/A8GRMZv4lViqAAAAAElFTkSuQmCC)
⇒ sin A cos C + cos A sin C = 1/4 + 3/4
⇒ sin A cos C + cos A sin C = 4/4 = 1
Thus, sin A cos C + cos A sin C = 1.
Question 10.In a right angle triangle ABC, right angle is at B, if
then find the value of
B. cos A cos C – sin A sin C
Answer:To find: cos A cos C – sin A sin C.
From previous part of the question, we have
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFgAAAAtCAMAAAAdpFx0AAAAAXNSR0IArs4c6QAAAJNQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOma2OpDbZgAAZgBmZjoAZjqQZpDbZrbbZrb/kDoAkDo6kDpmkGYAkGY6kLbbkNv/tmYAtmY6tpA6ttvbttv/tv//25A625Bm27Zm27aQ29u229v/2////7Zm/9uQ/9u2/9vb//+2///bjiFzMAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAByElEQVRYR+VWXUPCMAxs+KwgOFEYihSFCcJa1///60zbuQIOGDR7Mg8DHrhd75JLGfvflUBRpELo6ZwUrwDLnnb1AKfdenCZmDCm1z2ANq0k+mXFmGi8MRU3SJHlPUosjBwJIHe6Sh5+sUQTuVPU99agLPLzq9lgSYHK1AzM+bNHSzOLAAY0bfe1tUf3zbbhVFJIjvIm3rAUCrnDNNFxc7U/z/ZFJJXCxDYb07GBxJ8ksAavs7YktZkNFXdo3LMj0XMk1YwDDMlwscmoGuFYw+RkspUKbg/WGgc5UQYseWfJ1mQW+40QGXfzUb2atZoCNIa7FJCYeXxyGOYgIiRWsQN3+hWZSY79Inl/XqRpFoWsmizCxjZTkwNbdDeQZaqLYr9fPM0C+h8GxgPbd2FVjWt/mdj7hv/H7dfG1PsDbES62rHDP2w4ilkNuLoUbgN681DjQynEjQko+ZipETK2VtmHN++Z6feLJp2QS01xaHFHSfyY2IdZWo4ltjS0iPZXgFt6MyK/A7nuBBS1jvSs4w7kBaTbdkemVJ3Wa720U1ZDueVPXzoOydkzfEQ3NLXKwROThim9GPIO78f61sw6J4SLeXrGh+/8AWxnJOkm5mLuAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Using these values, we get
cos A cos C – sin A sin C = ![](data:image/png;base64,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)
⇒ cos A cos C – sin A sin C = √3/4 - √3/4 = 0
Thus, cos A cos C – sin A sin C = 0.
In right angle triangle ABC, 8 cm, 15 cm and 17 cm are the lengths of AB, BC and CA respectively. Then, find out sin A, cos A and tan A.
Answer:
We have
Given: ∠ABC = 90°, AB = 8 cm, BC = 15 cm and CA = 17 cm
We know sin A is given by,
⇒ [∵, perpendicular is the side opposite to the angle A & hypotenuse is the side opposite to the right angle of that triangle]
⇒ …(i)
Also, cos A is given by
⇒
⇒ …(ii)
Now, tan A can be found out by two ways:
Method 1: tan A is given by,
⇒
⇒
Method 2: tan A can also be written as,
⇒ [from equations (i) & (ii)]
⇒
Thus, ,
and
.
Question 2.
The sides of a right angle triangle PQR are PQ = 7 cm, QR = 25 cm and ∠Q = 90° respectively. Then find, tan P – tan R.
Answer:
We have
We don’t need to find hypotenuse (PR) in the ∆PQR as tan θ = perpendicular/base.
⇒
And
⇒
tan P – tan R =
⇒ tan P – tan R =
⇒ tan P – tan R = 576/175
Thus, tan P – tan R = 576/175
Question 3.
In a right angle triangle ABC with right angle at B, in which a = 24 units, b = 25 units and ∠BAC = θ. Then, find cosθ and tanθ.
Answer:
We have
In ∆ABC, ∠ABC = 90° and ∠BAC = θ.
Using this information, we can say
AC = hypotenuse of the triangle
BC = perpendicular (side opposite to the angle θ or ∠BAC)
Using Pythagoras theorem,
(hypotenuse)2 = (perpendicular)2 + (base)2
⇒ (25)2 = (24)2 + (base)2
⇒ (AB)2 = 625 – 576
⇒ (AB)2 = 49
⇒ AB = √49 = 7 units
So, we have AB = 7 units, BC = 24 units and AC = 25 units.
Thus,
⇒
And
⇒
Thus, and
.
Question 4.
Answer:
Given that,
But
⇒
⇒ base = 12 and hypotenuse = 13
So, using Pythagoras theorem, we can say
(hypotenuse)2 = (perpendicular)2 + (base)2
⇒ (perpendicular)2 = (hypotenuse)2 – (base)2
⇒ (perpendicular)2 = (13)2 – (12)2
⇒ (perpendicular)2 = 169 – 144 = 25
⇒ perpendicular = √25 = 5
Using perpendicular = 5, base = 12 and hypotenuse = 13, we can find out sin A and tan A.
Sin A is given by
⇒
And, tan A is given by
⇒
Thus, and
.
Question 5.
If 3 tanA = 4, then find sin A and cos A.
Answer:
Given that, 3 tan A = 4
⇒
But
⇒
⇒ perpendicular = 4 and base = 3
So, using Pythagoras theorem, we can say
(hypotenuse)2 = (perpendicular)2 + (base)2
⇒ (hypotenuse)2 = (4)2 + (3)2
⇒ (hypotenuse)2 = 16 + 9 = 25
⇒ hypotenuse = √25 = 5
Using perpendicular = 4, base = 3 and hypotenuse = 5, we can find out sin A and cos A.
Sin A is given by
⇒
And, cos A is given by
⇒
Thus, and
.
Question 6.
If ∠A and ∠X are acute angles such that cos A = cos X then show that ∠A = ∠X.
Answer:
We have
In this ∆AOX,
cos A = cos X
and cos A is given by,
[∵,
]
Similarly,
⇒
⇒ AO = OX [clearly, since denominator from either sides cancel each other]
Now, if sides AO and OX of ∆AOX are equal.
Then, ∠A = ∠X [∵, In a triangle, angles opposite to the equal sides are also equal]
Hence, ∠A = ∠X
Question 7.
Given then evaluate
A.
Answer:
We have been given that,
And we have to solve for .
We know the formula: (x + y)(x – y) = x2 – y2
Using this,
Also, we know the relationship between cos θ and sin θ which is given by
cos2 θ + sin2 θ = 1
⇒ cos2 θ = 1 – sin2 θ
So,
⇒
As we know,
So, B = 7 and P = 8
By Pythagoras theorem,
H2 = P2 + B2
H2 = 82 + 72
= 64 + 49
=113
H = √113
As we know,
So,
Question 8.
Given then evaluate
B.
Answer:
Given,
We know that, cosec2θ = 1 + cot2θ
Now, To Find:
Dividing both numerator and denominator by sin θ, we have
Question 9.
In a right angle triangle ABC, right angle is at B, if then find the value of
A. sin A cosC + cos A sin C
Answer:
We have
Given that, tan A = √3/1
And tan A is given by,
⇒
⇒ perpendicular = √3x and base = x
Then, we can use Pythagoras theorem in ∆ABC,
(hypotenuse)2 = (perpendicular)2 + (base)2
⇒ (hypotenuse)2 = (√3x)2 + (x)2
⇒ (hypotenuse)2 = 3x2 + x2 = 4x2
⇒ hypotenuse = √(4x2) = 2x
We have, AB = √3x, BC = x and AC = 2x.
Using these values,
⇒
⇒
⇒ …(i)
Similarly,
⇒
⇒ …(ii)
Also,
⇒
⇒
⇒ …(iii)
Similarly,
⇒
⇒ …(iv)
We have to solve: sin A cos C + cos A sin C.
Substituting equations (i), (ii), (iii) & (iv) in above,
sin A cos C + cos A sin C =
⇒ sin A cos C + cos A sin C = 1/4 + 3/4
⇒ sin A cos C + cos A sin C = 4/4 = 1
Thus, sin A cos C + cos A sin C = 1.
Question 10.
In a right angle triangle ABC, right angle is at B, if then find the value of
B. cos A cos C – sin A sin C
Answer:
To find: cos A cos C – sin A sin C.
From previous part of the question, we have
Using these values, we get
cos A cos C – sin A sin C =
⇒ cos A cos C – sin A sin C = √3/4 - √3/4 = 0
Thus, cos A cos C – sin A sin C = 0.
Exercise 11.2
Question 1.Evaluate the following.
A. sin 45° + cos 45°
Answer:By trigonometric identities, we can say
sin 45° = 1/√2
and cos 45° = 1/√2
Adding them, we get
sin 45° + cos 45° = 1/√2 + 1/√2
⇒ sin 45° + cos 45° = 2/√2 = √2
Thus, sin 45° + cos 45° = √2.
Question 2.Evaluate the following.
B. ![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
Trigonometric identities:
cos 45° = 1/√2
sec 30° = 1/cos 30° = 2/√3 [∵, cos 30° = √3/2]
cosec 60° = 1/sin 60° = 2/√3 [∵, sin 60° = √3/2]
Putting the values we get,![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 3.Evaluate the following.
C. ![](data:image/png;base64,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)
Answer:By trigonometric identities,
cot 45° = 1/tan 45° = 1/1 = 1
sec 30° = 1/cos 30° = 2/√3 [∵, cos 30° = √3/2]
cosec 60° = 1/sin 60° = 2/√3 [∵, sin 60° = √3/2]
sin 30° = 1/2
tan 45° = 1
cos 60° = 1/2
Putting all these values in
, we get
![](data:image/png;base64,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)
Since, numerator is equal to denominator in the above calculation, we can say
![](data:image/png;base64,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)
Question 4.Evaluate the following.
D. 2tan2 45° + cos2 30° - sin2 60°
Answer:By trigonometric identities,
sin 60° = √3/2
tan 45° = 1
cos 30° = √3/2
Putting these values in 2 tan2 45° + cos2 30° - sin2 60°, we get
2 tan2 45° + cos2 30° - sin2 60° = 2(1)2 + (√3/2)2 – (√3/2)2
⇒ 2 tan2 45° + cos2 30° - sin2 60° = 2
Thus, 2 tan2 45° + cos2 30° - sin2 60° = 2.
Question 5.Evaluate the following.
E. ![](data:image/png;base64,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)
Answer:We have to solve:
![](data:image/png;base64,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)
Recall the trigonometric identities,
sin2 θ + cos2 θ = 1
& sec2 α – tan2 α = 1
Put θ = 30° and α = 60°, we get
sin2 30° + cos2 30° = 1
& sec2 60° - tan2 60° = 1
So,
![](data:image/png;base64,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)
Thus,
![](data:image/png;base64,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)
Question 6.Choose the right option and justify your choice -
A) ![](data:image/png;base64,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)
A. sin 60°
B. cos 60°
C. tan 30°
D. sin 30°
Answer:we know,
tan 30° = 1/√3
tan 45° = 1
Then, putting these values in the question, we get
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALgAAAAqCAMAAAA3Q2G9AAAAAXNSR0IArs4c6QAAAMlQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmZmOmaQOma2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZjqQZmaQZma2ZpDbZra2ZrbbZrb/kDoAkDo6kGY6kGaQkJCQkJC2kLbbkNv/tmYAtmY6tmZmtpA6tpBmtpC2ttvbttv/tv/btv//25A625Bm25CQ27Zm27aQ27a229uQ29u229vb2/+22////7Zm/9uQ/9u2//+2///bTDq3vwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADfklEQVRYR+1ZDVfaMBRNqjLr50QQt+mw4CbFsTmKuLk1Jfn/P8r3krQN0o0kosDRnFMobT5uXu57eTcQ8lY8LSDGu0PPpkttJr7fn6wlcEL4G/AXpk62phaPKaXbL2wrh+HuW5TWHeqvSlXevCLjoCfh9C/gg39MVwXbXBzZjgLOTy/gUvdrUZJjDZOffgbsSysiokcuyz0uKZ6Feg7gl2XxnknpP0mll4/2aHAOvYuu/GZtkjjYLSlxP8Hioj8LzfCfuAq46Gj/SrZTpKsbcAa4mZon8ht57lFG71vVkVP5zzdYvlo66VJ6TBitDcIgHwXf88YFEdExXLRuTRXeQEqobq5xDDOqYCSHYjWRajIQ7T9ocd5ok34tJUlwRWL4xjLptuFFEwauXBOrkWcr8TOHhtXAc//JYTEEri/oPKabzwFcM8gOfSXwwn8k8PE+EsYETshdCCFYU8VuIItaox64O92yy22rgJf+A9T4w2CfS6C3wuJIkQyAI3HyvcQC1twq4jLlJze8ZeWtIn4341eG/2RhcM4jujWg21koL4y7o1CFQ96iAVBmEeUW4GbYJ7k7tHH0BJ0438cWAcC7j+wwHYGvg9GcNjPv8RbXMG5fphgbMcCuVWH7wBQgn61zrszkRGdV0kS9DYIT6fTb3UZ2XUwlaN4/5qGz6HhOF497mDfi63lvt87/tccCung99n72mcq90zoJLuCIzrID7rVdLvbYgDIbmtniXNXnE5ZlKnev0nHQN6sWSfyDnLPME3HdIFq7iThs7X08xGCfzbXvlI4TUZ5d/iN5yD7JLE509hC4quwM3DwecjM/ZGSF9oXxS9sWwGWSC6QYh3TzYCiioNdHWcHOUpmGJm2coQbupD5zoF4pPeTusr1uXAWcZPJ/AgaKE17f/ogOevAknw6ryxkiVXwPBIvjIUuTiwEYLFPyVGpfNJ+2uJISWoyj3gQlD/BiMDzqNflEFuC5iDAf1Vy3HNusZhwP2bX+NcAFl9xU2tcADrclxxGuXBKp6tnGUE5AFRVM5Tw882jjeMgOt4KWB0PUvvDE0HEFcLiZAJ93epPu7s+vuCjw5LYMolgRU1IZXZyLcTxk3zap/UWmFNp3SseVFu9jag/y++i+sXUjD4BiU2n+DjeGog/R0Ed9msdD9sB584uUp7n2tW+57JqJd4a/ZOSgrEsED1Tiap33FedTAAAAAElFTkSuQmCC)
And we know, tan 30° = 1/√3
But, sin 60° = √3/2
cos 60° = 1/2
& sin 30° = 1/2
Thus, option (C) is correct.
Question 7.Choose the right option and justify your choice -
B) ![](data:image/png;base64,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)
A. tan 90°
B. 1
C. sin 45°
D. 0
Answer:We know that,
tan 45° = 1
So, using this value of tangent, we can write
![](data:image/png;base64,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)
The answer has come out to be 0.
Thus, option (D) is correct.
Question 8.Choose the right option and justify your choice -
C) ![](data:image/png;base64,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)
A. cos 60°
B. sin 60°
C. tan 60°
D. sin 30°
Answer:We know that,
tan 30° = 1/√3
So, using this value of tangent, we can write
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHsAAAAxCAMAAADN2iZHAAAAAXNSR0IArs4c6QAAALRQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmZmOmaQOma2OpDbZgAAZgA6ZgBmZjoAZjpmZjqQZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGY6kGaQkJC2kLbbkNv/tmYAtmY6tpA6tpBmtpC2ttvbttv/tv/btv//25A625Bm25CQ27Zm27aQ29u229vb2////7Zm/9uQ/9u2//+2///b2p5KUwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACo0lEQVRYR+1XDXObMAy1aZOFdv1Om21tQtIVsm5dIGmGif3//9ckQcJHnAacHt5u8x3HYYOfnpCsJ8b+iaHGnHdmdqjKuxc5uLGDDajykyXejCVfYlu0xX2cPNgBl33Oub3/bYf0f9SDPaA8ftUoXyDM8lGFXw44v8bJsKezLPrIHcwPNaG7GLJwdDCDbAN5+8QWjg9PgQ5bjbPlsBcnJ/77YqMJuCn7Bq7pxqsJpqLg3WfXWfPDZdkfMeXdwMWvG/k8pRikftc4LKTER96yP2TTbsxC54kFcMexmgxh4RbM03qmzg+Q97veWtDv3uwsEDu7yOTjg7HFrhAJU+gUe3GGni9iM/bqjjY+r8Ny+53Ih1DV1HAB0GQX+PiXgKgLoc5veKOvE8DGVQoLk6EeY6rhVfZ00HOcTVznQXq888x7iUsXhkHkpjkmB9wB3zcec0BMi9jrhUGQNsYrfZBcxBH4Czg2O5QOQ82+DoaPMZOfY0zRtoc4A5fDL7OhF9XYMERrOSk7tSBm6XDeP0oFxuRhD8QbW+437u94o7bP1eLUlk5X35d3trAh1Rphv28j1wxb38g1lm1Z4CZr3mpaUUlCq9jSRq6s2wxlGwYlYUSXgzLWjgOXGrmKbquJvcUuz9uyMl23TQuXH5/PlOf4U5QReSNX0G31ZNsWu8KRUVHFCSWfAKkGC/Mf3rkPM3kjZ6Lb9LobTaCVVDygfEChBkIURQpUO1SmNLMuhSa6bQ927gdEJL1MolQczciGfBjotjK7glBWwYeiigHGK2jST/zV5PTnV3QKzMyzc8hQt+3iHaIZRWJTrO4gKK+W/c4LNQJBLtGMdFuFXZvVaYtdm+B/EJa+QWjHQG2D0A40olhoEDJydhoEArfUIKTYdhqEUkT9BlodUYjt+EL5AAAAAElFTkSuQmCC)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHgAAAAxCAMAAAAm7Z1EAAAAAXNSR0IArs4c6QAAALRQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmZmOmaQOma2OpDbZgAAZgA6ZgBmZjoAZjpmZjqQZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGY6kGaQkJC2kLbbkNv/tmYAtmY6tpA6tpBmtpC2ttvbttv/tv/btv//25A625Bm25CQ27Zm27aQ29u229vb2////7Zm/9uQ/9u2//+2///b2p5KUwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACoElEQVRYR+2XDVOjMBCGE872ip56arV62tJ6Qs9TC8Ueocn//1+3m9Dy0XQk4JDpzWWG6TQB3t1ls/uEkH9+iCmlvYUFN/nNKx9dWRAGSX5rw2NC0h+JFYfZfZI+WFDmQ0qppW9swd3/ko0jIDx6YbI3IK/yUVZdjSi9xKlwoDMn+kYd3AtiJn/ZmISTxmYXH+TXTyR2fJgKdMJimi2HgyQ99j9RGI3AN5JfEJJ+sp7hnmO0/+w6G89wmQ8nRHhXcNFLk1ArHwMV7p1IhXJ7o8d8OCbzfkJC54kE8ItjPRvDwjXYpo1Jjbjze/1NsfzE29cyFM4uae1RW2Gmz4pQ6Srh+BQDXhQm5N2dbENdw7/dWyIfUnOnJzPQlSZBaP8wSLMQuvbWYwxxCsK4KlOhwRCPiezJZb9lCac4l7rOA/do75kOUlde+OkjV20nPqIOhNx0LEFOtab3c/OkNFUr3p+eJxEECvwzKjxtJLNng/FjQvhdgpux08FOIdLwoToHPjFtlpJ1opMVJkhSWXs/GKW+0eRPw/d/ZNdBrO8JdbVcifikG7yulCvxe3XTjXBWrqAEZsWRmwi3OXJVy5WRsP7IVQu9dspVmnks5hXYYVrqUkeuMnvVQq9qucIcRIXo+6gstKeayiNXhb1qCe/fkWWu3JxxYpcenS2E5/hzZIL8yFVgr2botbWkArSp3GYMcAsWli/emQ8z+ZGrPXuVhRUJIAsgbAFGIm5AB0OulDOb9taevfZ5jHISdSVSsi8LaUA+TNlL2wDwdSL4WqQR8HUNR+hjfz07efuJ4YaZZVZrPoO9Ni6EaFHRpTn2aiiwF6th71XSe5BjVkv2Ooiec/BG6mC+E6c0MN+JLop0DfOZYxZgXirbgHklbAHmdUn0F50NUwND93dTAAAAAElFTkSuQmCC)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
And we know, tan 60° = √3
But, cos 60° = 1/2
sin 60° = √3/2
& sin 30° = 1/2
Thus, option (C) is correct.
Question 9.Evaluate sin 60° cos 30° + sin 30°cos 60°. What is the value of sin(60° + 30°). What can you conclude?
Answer:Let us first solve sin 60° cos 30° + sin 30° cos 60°.
We know,
sin 60° = √3/2
cos 30° = √3/2
sin 30° = 1/2
& cos 60° = 1/2
So, sin 60° cos 30° + sin 30° cos 60° = ![](data:image/png;base64,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)
⇒ sin 60° cos 30° + sin 30° cos 60° = ![](data:image/png;base64,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)
⇒ sin 60° cos 30° + sin 30° cos 60° = 1 …(i)
Now, for sin (60° + 30°):
sin (60° + 30°) = sin 90°
⇒ sin (60° + 30°) = 1 [∵, sin 90° = 1] …(ii)
By equations (i) & (ii), we can conclude that
sin (60° + 30°) = sin 60° cos 30° + sin 30° cos 60°
And infact in general, let 60° = x and 30° = y. Then,
sin (x + y) = sin x cos y + sin y cos x
Question 10.Is it right to say cos(60° + 30°) = cos 60° cos 30° - sin 60° sin 30°.
Answer:Let us solve Left-Hand-Side:
cos (60° + 30°) = cos 90°
⇒ cos (60° + 30°) = 0 [∵, cos 90° = 0]
Now, solve for Right-Hand-Side:
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ cos 60° cos 30° - sin 60° sin 30° = 0
Question 11.In right angle triangle ΔPQR, right angle is at Q and PQ = 6cms ∠RPQ = 60°. Determine the lengths of QR and PR.
Answer:![](data:image/png;base64,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)
To find QR:
Since, tan θ = perpendicular/base
We know that,
![](data:image/png;base64,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)
⇒
[∵, PQ = 6 cm & tan 60° = √3]
⇒ QR = 6√3
Now, PR can be found by two ways -
1st method: In ∆PQR, using Pythagoras theorem,
PR2 = PQ2 + QR2 [∵, (hypotenuse)2 = (perpendicular)2 + (base)2]
⇒ PR2 = 62 + (6√3)2
⇒ PR2 = 36 + 108 = 144
⇒ PR = √144 = 12
2nd Method:
Since, cos θ = base/hypotenuse
We know that,
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ PR = 2 × PQ
⇒ PR = 2 × 6 = 12
Thus, QR = 6√3 cm and PR = 12 cm.
Question 12.In right angle is at Y, YZ = x and XZ = 2x then determine ∠YXZ and ∠YZX.
Answer:We have
![](data:image/png;base64,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)
With given values, YZ = x and XZ = 2x, we can find out both angles.
For ∠YXZ:
Let ∠YXZ = θ, then
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ θ = sin-1(1/2)
⇒ θ = 30° [∵, sin 30° = 1/2]
For ∠YZX = α, then
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ α = cos-1(1/2)
⇒ α = 60° [∵, cos 60° = 1/2]
Question 13.Is it right to say that sin(A + B) = Sin A + Sin B? Justify your answer.
Answer:No, it is not correct to say that sin (A + B) = sin A + sin B.
Justification: Let’s justify it by showing contradiction.
Let it be true that, sin (A + B) = sin A + sin B.
Now, let A = 30° and B = 60°
Then,
sin (30° + 60°) = sin 30° + sin 60°
⇒ sin 90° = sin 30° + sin 60°
⇒ 1 = 1/2 + √3/2
⇒ 1 = (1 + √3)/2
But, it’s not true.
1 ≠ (1 + √3)/2
Hence, we have a contradiction.
And therefore, it’s not right to say that sin (A + B) = sin A + sin B.
Evaluate the following.
A. sin 45° + cos 45°
Answer:
By trigonometric identities, we can say
sin 45° = 1/√2
and cos 45° = 1/√2
Adding them, we get
sin 45° + cos 45° = 1/√2 + 1/√2
⇒ sin 45° + cos 45° = 2/√2 = √2
Thus, sin 45° + cos 45° = √2.
Question 2.
Evaluate the following.
B.
Answer:
Trigonometric identities:
cos 45° = 1/√2
sec 30° = 1/cos 30° = 2/√3 [∵, cos 30° = √3/2]
cosec 60° = 1/sin 60° = 2/√3 [∵, sin 60° = √3/2]
Putting the values we get,Question 3.
Evaluate the following.
C.
Answer:
By trigonometric identities,
cot 45° = 1/tan 45° = 1/1 = 1
sec 30° = 1/cos 30° = 2/√3 [∵, cos 30° = √3/2]
cosec 60° = 1/sin 60° = 2/√3 [∵, sin 60° = √3/2]
sin 30° = 1/2
tan 45° = 1
cos 60° = 1/2
Putting all these values in , we get
Since, numerator is equal to denominator in the above calculation, we can say
Question 4.
Evaluate the following.
D. 2tan2 45° + cos2 30° - sin2 60°
Answer:
By trigonometric identities,
sin 60° = √3/2
tan 45° = 1
cos 30° = √3/2
Putting these values in 2 tan2 45° + cos2 30° - sin2 60°, we get
2 tan2 45° + cos2 30° - sin2 60° = 2(1)2 + (√3/2)2 – (√3/2)2
⇒ 2 tan2 45° + cos2 30° - sin2 60° = 2
Thus, 2 tan2 45° + cos2 30° - sin2 60° = 2.
Question 5.
Evaluate the following.
E.
Answer:
We have to solve:
Recall the trigonometric identities,
sin2 θ + cos2 θ = 1
& sec2 α – tan2 α = 1
Put θ = 30° and α = 60°, we get
sin2 30° + cos2 30° = 1
& sec2 60° - tan2 60° = 1
So,
Thus,
Question 6.
Choose the right option and justify your choice -
A)
A. sin 60°
B. cos 60°
C. tan 30°
D. sin 30°
Answer:
we know,
tan 30° = 1/√3
tan 45° = 1
Then, putting these values in the question, we get
⇒
And we know, tan 30° = 1/√3
But, sin 60° = √3/2
cos 60° = 1/2
& sin 30° = 1/2
Thus, option (C) is correct.
Question 7.
Choose the right option and justify your choice -
B)
A. tan 90°
B. 1
C. sin 45°
D. 0
Answer:
We know that,
tan 45° = 1
So, using this value of tangent, we can write
The answer has come out to be 0.
Thus, option (D) is correct.
Question 8.
Choose the right option and justify your choice -
C)
A. cos 60°
B. sin 60°
C. tan 60°
D. sin 30°
Answer:
We know that,
tan 30° = 1/√3
So, using this value of tangent, we can write
⇒
⇒
⇒
⇒
⇒
And we know, tan 60° = √3
But, cos 60° = 1/2
sin 60° = √3/2
& sin 30° = 1/2
Thus, option (C) is correct.
Question 9.
Evaluate sin 60° cos 30° + sin 30°cos 60°. What is the value of sin(60° + 30°). What can you conclude?
Answer:
Let us first solve sin 60° cos 30° + sin 30° cos 60°.
We know,
sin 60° = √3/2
cos 30° = √3/2
sin 30° = 1/2
& cos 60° = 1/2
So, sin 60° cos 30° + sin 30° cos 60° =
⇒ sin 60° cos 30° + sin 30° cos 60° =
⇒ sin 60° cos 30° + sin 30° cos 60° = 1 …(i)
Now, for sin (60° + 30°):
sin (60° + 30°) = sin 90°
⇒ sin (60° + 30°) = 1 [∵, sin 90° = 1] …(ii)
By equations (i) & (ii), we can conclude that
sin (60° + 30°) = sin 60° cos 30° + sin 30° cos 60°
And infact in general, let 60° = x and 30° = y. Then,
sin (x + y) = sin x cos y + sin y cos x
Question 10.
Is it right to say cos(60° + 30°) = cos 60° cos 30° - sin 60° sin 30°.
Answer:
Let us solve Left-Hand-Side:
cos (60° + 30°) = cos 90°
⇒ cos (60° + 30°) = 0 [∵, cos 90° = 0]
Now, solve for Right-Hand-Side:
⇒
⇒ cos 60° cos 30° - sin 60° sin 30° = 0
Question 11.
In right angle triangle ΔPQR, right angle is at Q and PQ = 6cms ∠RPQ = 60°. Determine the lengths of QR and PR.
Answer:
To find QR:
Since, tan θ = perpendicular/base
We know that,
⇒ [∵, PQ = 6 cm & tan 60° = √3]
⇒ QR = 6√3
Now, PR can be found by two ways -
1st method: In ∆PQR, using Pythagoras theorem,
PR2 = PQ2 + QR2 [∵, (hypotenuse)2 = (perpendicular)2 + (base)2]
⇒ PR2 = 62 + (6√3)2
⇒ PR2 = 36 + 108 = 144
⇒ PR = √144 = 12
2nd Method:
Since, cos θ = base/hypotenuse
We know that,
⇒
⇒
⇒ PR = 2 × PQ
⇒ PR = 2 × 6 = 12
Thus, QR = 6√3 cm and PR = 12 cm.
Question 12.
In right angle is at Y, YZ = x and XZ = 2x then determine ∠YXZ and ∠YZX.
Answer:
We have
With given values, YZ = x and XZ = 2x, we can find out both angles.
For ∠YXZ:
Let ∠YXZ = θ, then
⇒
⇒
⇒ θ = sin-1(1/2)
⇒ θ = 30° [∵, sin 30° = 1/2]
For ∠YZX = α, then
⇒
⇒
⇒ α = cos-1(1/2)
⇒ α = 60° [∵, cos 60° = 1/2]
Question 13.
Is it right to say that sin(A + B) = Sin A + Sin B? Justify your answer.
Answer:
No, it is not correct to say that sin (A + B) = sin A + sin B.
Justification: Let’s justify it by showing contradiction.
Let it be true that, sin (A + B) = sin A + sin B.
Now, let A = 30° and B = 60°
Then,
sin (30° + 60°) = sin 30° + sin 60°
⇒ sin 90° = sin 30° + sin 60°
⇒ 1 = 1/2 + √3/2
⇒ 1 = (1 + √3)/2
But, it’s not true.
1 ≠ (1 + √3)/2
Hence, we have a contradiction.
And therefore, it’s not right to say that sin (A + B) = sin A + sin B.
Exercise 11.3
Question 1.Evaluate
A) ![](data:image/png;base64,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)
Answer:By trigonometric identity, we have
tan (90° - θ) = cot θ
Replace θ = 54°
⇒ tan (90° - 54°) = cot 54°
⇒ tan 36° = cot 54° [∵, 90° - 54° = 36°]
Now, ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Question 2.Evaluate
B) cos 12° - sin 78°
Answer:By trigonometric identity, we can say
cos (90° - θ) = sin θ
Now, just replace θ by 78°.
We get
cos (90° - 78°) = sin 78°
⇒ cos 12° = sin 78° [∵, 90° - 78° = 12°]
Now, cos 12° - sin 78° = sin 78° - sin 78°
⇒ cos 12° - sin 78° = 0
Question 3.Evaluate
C) cosec 31° - sec 59°
Answer:By trigonometric identity, we can say
cosec (90° - θ) = sec θ
Replace θ by 59°.
We get
cosec (90° - 59°) = sec 59°
⇒ cosec 31° = sec 59°
Now, cosec 31° - sec 59° = sec 59° - sec 59°
⇒ cosec 31° - sec 59° = 0
Question 4.Evaluate
D) sin 15° sec 75°
Answer:By trigonometric identities, we can say
sin (90° - θ) = cos θ
And sec θ = 1/cos θ
Replace θ by 75°.
We get
sin (90° - 75°) = cos 75°
⇒ sin 15° = cos 75° [∵, 90° - 75° = 15°]
And sec 75° = 1/cos 75°
Using these values, we can solve the given expression.
sin 15° sec 75° = sin 15°/cos 75°
⇒ sin 15° sec 75° = cos 75°/cos 75°
⇒ sin 15° sec 75° = 1
Question 5.Evaluate
E. tan 26° tan 64°
Answer:By trigonometric identities, we can say
tan (90° - θ) = cot θ
And tan θ = 1/cot θ
Replace θ by 64°.
We get
tan (90° - 64°) = cot 64°
⇒ tan 26° = cot 64° [∵, 90° - 64° = 26°]
And tan 64° = 1/cot 64°
Using these values, we can solve for the given expression.
tan 26° tan 64° = cot 64° tan 64°
⇒ tan 26° tan 64° = cot 64°/cot 64°
⇒ tan 26° tan 64° = 1
Question 6.Show that
A. tan 48° tan16° tan 42° tan 74° = 1
Answer:We have
LHS = tan 48° tan 16° tan 42° tan 74° = (tan 48° tan 42°)(tan 16° tan 74°)
We know the trigonometric identities, we can say
tan (90° - θ) = cot θ
And tan θ = 1/cot θ
First, replace θ by 42°.
tan (90° - 42°) = cot 42°
⇒ tan 48° = cot 42° …(1)
And tan 42° = 1/cot 42° …(2)
Now, replace θ by 74°.
tan (90° - 74°) = cot 74°
⇒ tan 16° = cot 74° …(3)
And tan 74° = 1/cot 74° …(4)
Using equations (1), (2), (3) & (4), we get
LHS = tan 48° tan 16° tan 42° tan 74° = (tan 48° tan 42°)(tan 16° tan 74°)
= (cot 42°/cot 42°)(cot 74°/cot 74°)
= 1 = RHS
Thus, tan 48° tan 16° tan 42° tan 74° = 1.
Question 7.Show that
B. cos36°cos54° - sin36°sin54°
Answer:We have
LHS = cos 36° cos 54° - sin 36° sin 54°
We know the trigonometric identity,
cos (90° - θ) = sin θ
First, replace θ by 54°.
We get, cos (90° - 54°) = sin 54°
⇒ cos 36° = sin 54° …(1)
Now, replace θ by 36°.
We get, cos (90° - 36°) = sin 36°
⇒ cos 54° = sin 36° …(2)
Using equations (1) & (2), we get
LHS = cos 36° cos 54° - sin 36° sin 54° = sin 54° sin 36° - sin 54° sin 36°
= 0 = RHS
Thus, cos 36° cos 54° - sin 36° sin 54° = 0.
Question 8.If tan2A = cot(A – 18°) where 2A is an acute angle. Find the value of A.
Answer:Given that, 2A is an acute angle.
⇒ 2A < 90°
So, using trigonometric identity, we can say that
cot (90° - 2A) = tan 2A [∵, cot (90° - θ) = tan θ]
Now, replace tan 2A by cot (90° - 2A) in the given question.
tan 2A = cot (A – 18°)
⇒ cot (90° - 2A) = cot (A – 18°)
Now, we can compare the degrees from above, we get
90° - 2A = A – 18°
⇒ 2A + A = 90° + 18°
⇒ 3A = 108°
⇒ A = 108°/3
⇒ A = 36°
Thus, the value of A is 36°.
Question 9.If tan A = cot B where A and B are acute angles, prove that A + B = 90°
Answer:Given that, A and B are acute angles.
⇒ A < 90° & B < 90°
So, using trigonometric identity, we can say
tan (90° - B) = cot B [∵, tan (90° - θ) = cot θ]
Replace cot B of RHS by tan (90° - B) in the given question.
tan A = cot B
⇒ tan A = tan (90° - B)
Now, comparing the degrees from the above, we get
A = 90° - B
⇒ A + B = 90°
Hence, proved that A + B = 90°.
Question 10.If A, B and C are interior angles of a triangle ABC, then show that ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALAAAAA7CAYAAADbwABFAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAddSURBVHja7Z0/T9xIFMDnGpokdQJiIqUgSkp2WKU6pUGbOUKKFEgx1ioFaIsVspQL0hboZOiQjnyAS5V0l+JariBlmvANIkGXbvkMbHZmduxZr//M2l57bL+TXnFAwH7vN2/ev5lFJycnCASkqgJKAAGAQUAAYBCQugA8/u83x6ZblNp7vdPTu3VV/ulp765N6R61nS32zgBkDQAeDHr3dgn+0LadN00xgmO332Cy+6E3GNwDKCsMsISX2Mfbabx2GV5M/t0kSfo9xzbZZhA3FsQUOjMOYAYv7hy8T/dv0We0Sr8WGXKwEIBidDFW9u1YRhO5nRKMf3Dv2nOXk37fQQe/T/v+VQb3qL/z+AV5+BEjNJR6w3jjfKffbx31N1uE2K7xAHMPhOmXNAC6bneZIHSNEL6hjrte9LOzv9/C6BIhct11fVCPHeeBTfCZMAq+2ewftZIXBP6SZgeqosgdlwNL7DPHOX7ghVWOtU4w+s6dAtn9bDTA0nBp4WOeS3rBMjwYf/5V9DUIcOD5Rjo7hOvQ9bQLuWrCd80Ym3k7nOkAM+8b95A68JBud5974RzCiHm38iSAOZQI3UR9P8ywdffC3OYaizppQZvhfRmAKQ2mvqBY0dnDiNwBtsgrXQ/sGbfgeL6cHQuNkuzOfrZDOn8ZC7DwTq1LHc8U5a0kbN6qTunNcwF4MFhRM+kDq20J74uHuouUx9QIX5YRzxeVM+SVs1Q6fBCGbl1I+OfdqvMHOKQKwYVcswRunjJfncMI3041ADiLoXhyNIZfejuW1epuTcFSzpTX7ODDMcCHunXJqBCCfZ03KUR5aDRPnTfLwq4OwNkcTekAZ6k+KF4vxOPphxFKBSNRohaFfhKnv7DqXI2ojQcOhgBzhx4hiU4eqzvvJE4tGekurCy6qVMSV1uAo0KPPJQDAC9evNq4hj7YwFOUHsrfSlJsk6y9GFe5yFqNyBvgwaC7IrJu/UZL1uZOJbywaMOPZL4Rlps4HbIf54gqB7CXqG3Q/6KmtwTgIuZsWwfWvEM+8wAsksfuimwl7/SPHk+X0TovxffYXAS90H3XugMsd5lJu/j2IXnxUdWd2+st68zGVArgSf3wyk+s8DBo4In3nU7AiP1pEQAHhnkiBWP8jY2HzrNQmwCwFDb7vYHx+bTONs4tx10zOwZuUN+/yApNkyQSLNJx9hcO8Nsnr5eWnl6N48b7YIyZ3eb+06Wlqydv3deVe/aC+IkEWG3PAsAAcJoKUVFTgTMJCetCqZmhmgDpdKaifh4ANhtg3a5j3M8l8bNQgFnM9QfG/yqnC6YSIJb9s6l5JYG6WaW7x2p1wKLkT4zRD1b2YIG5zDD5iYSQOBdiYDNiYHYiYpxE/S9tyyoCwQoPs69N2+/8UxN4qJ7jS+KnsBBC1lDVLUCtc8rM0KbYVZsFNkGf5MMzBbCTtjKjj2oqAMDZAM7jPJ4sOcpDtF4nU+lyevZXyoBsxoPbO9ANDePHDIBZuUh5ATnjqv5cGKxSIWEvBACnB3ieGY7YEw8MzEDDRxyDIleyKSP+1uzcgrS3+vuNBDi4lUxNWGkCHNYVA4DLDSGkbeI6XR7kcXMnyveMBVgZzh7yA3gdsl80wLpbZhXEBIClreMA9obPwwD2BtP9FnoSwHnrTRtgcXrU30bCfq4AgG/rIibEwDreMg7gsBkQDYBz1ZvWS4VBWAbAEELkGwNPjZ4OBiuxIURIDOzBncBF6SGE7tcA4GrFwFNHocb2YSUx6bWZfaUdlWTtMKyCodq7dIBV4ETtj+45zubvImkTK9V1rLXJ8AV/cXYJBZskUs75ey86dSI3UFsEgMuvAyvDTyNf8I1acVJHH+V0nzcVGKjzhvGzSPuGftG7TQaT77Luy5I2ATEesj63Pxk2/v9udz84lcVW4Mw2FxgnBIDNaGTw2YVJ04k3MjboP8FJMH6TpnfLkGhktKn9Lsx2YfwUCnBRAq3kGN1UeBaiSAGAAWAAuA4Ap7naEwAuX/8A8IkYZpkc+xFJDG5ddqyDlwCw+fpvPMDK+bnpeyVYOdByXwHAZuu/0QDLOiimtusNKTnWmiwZ5XFzDAC8WP03GuCooy9x3ScA2Cz9l/wCz57fQfjntus+Mk25oilTpgfefoTRnZ/PHPd5E6sLuvovO9M28vYZf052sacJqqgb0/Rf/oMaeHQ8653FuT1DQ7uU8+jfiK3CtHtw037UV54ir45tZvigr//SH9a0e3BN+airJnxORh76N2O7MGSr5COEBV3IkRz/1vcjBvLUvzEBe9nehpd0Qj5Qr4y2ct0/5CVP/Ruz8soMI9jMMrvjINiPZy1OQujfRYPUtPAhi/7NKZuUVI3wP0VIHej2peh4uGnVh6z6Nyr+Kdpw3uUckTJ7fWsRC7kp3jcP/Rv1QmySv2kfdj2TgRP7rKnvn0aMehh5bKWJ5SN+WcwYXjheVWGAVYjlXV0AL0ilAPaNSrcotffqbFS+WCndkxchgtQIYBAQABgEAAYBMVl+ATJc8C+I1O6qAAAAAElFTkSuQmCC)
Answer:If A, B and C are interior angles of ∆ABC, then we can say that
A + B + C = 180° (by angle sum property of a triangle)
⇒ A + B = 180° - C
Take LHS:
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
[∵, tan (90° - θ) = cot θ, where θ = C/2 here]
Hence,
.
Question 11.Express sin 75° + cos 65° in terms of trigonometric ratios of angles between 0o and 45o.
Answer:Given: sin 75° + cos 65°.
We can write,
75° = 90° - 15°
& 65° = 90° - 25°
Then, sin 75° + cos 65° = sin (90° - 15°) + cos (90° - 25°)
⇒ sin 75° + cos 65° = cos 15° + sin 25°
[∵, sin (90° - θ ) = cos θ & cos (90° - θ) = sin θ]
In cos 15° + sin 25°, 15° & 25° both are angles between 0° and 45°.
Thus, answer is sin 75° + cos 65° = cos 15° + sin 25°.
Evaluate
A)
Answer:
By trigonometric identity, we have
tan (90° - θ) = cot θ
Replace θ = 54°
⇒ tan (90° - 54°) = cot 54°
⇒ tan 36° = cot 54° [∵, 90° - 54° = 36°]
Now,
⇒
Question 2.
Evaluate
B) cos 12° - sin 78°
Answer:
By trigonometric identity, we can say
cos (90° - θ) = sin θ
Now, just replace θ by 78°.
We get
cos (90° - 78°) = sin 78°
⇒ cos 12° = sin 78° [∵, 90° - 78° = 12°]
Now, cos 12° - sin 78° = sin 78° - sin 78°
⇒ cos 12° - sin 78° = 0
Question 3.
Evaluate
C) cosec 31° - sec 59°
Answer:
By trigonometric identity, we can say
cosec (90° - θ) = sec θ
Replace θ by 59°.
We get
cosec (90° - 59°) = sec 59°
⇒ cosec 31° = sec 59°
Now, cosec 31° - sec 59° = sec 59° - sec 59°
⇒ cosec 31° - sec 59° = 0
Question 4.
Evaluate
D) sin 15° sec 75°
Answer:
By trigonometric identities, we can say
sin (90° - θ) = cos θ
And sec θ = 1/cos θ
Replace θ by 75°.
We get
sin (90° - 75°) = cos 75°
⇒ sin 15° = cos 75° [∵, 90° - 75° = 15°]
And sec 75° = 1/cos 75°
Using these values, we can solve the given expression.
sin 15° sec 75° = sin 15°/cos 75°
⇒ sin 15° sec 75° = cos 75°/cos 75°
⇒ sin 15° sec 75° = 1
Question 5.
Evaluate
E. tan 26° tan 64°
Answer:
By trigonometric identities, we can say
tan (90° - θ) = cot θ
And tan θ = 1/cot θ
Replace θ by 64°.
We get
tan (90° - 64°) = cot 64°
⇒ tan 26° = cot 64° [∵, 90° - 64° = 26°]
And tan 64° = 1/cot 64°
Using these values, we can solve for the given expression.
tan 26° tan 64° = cot 64° tan 64°
⇒ tan 26° tan 64° = cot 64°/cot 64°
⇒ tan 26° tan 64° = 1
Question 6.
Show that
A. tan 48° tan16° tan 42° tan 74° = 1
Answer:
We have
LHS = tan 48° tan 16° tan 42° tan 74° = (tan 48° tan 42°)(tan 16° tan 74°)
We know the trigonometric identities, we can say
tan (90° - θ) = cot θ
And tan θ = 1/cot θ
First, replace θ by 42°.
tan (90° - 42°) = cot 42°
⇒ tan 48° = cot 42° …(1)
And tan 42° = 1/cot 42° …(2)
Now, replace θ by 74°.
tan (90° - 74°) = cot 74°
⇒ tan 16° = cot 74° …(3)
And tan 74° = 1/cot 74° …(4)
Using equations (1), (2), (3) & (4), we get
LHS = tan 48° tan 16° tan 42° tan 74° = (tan 48° tan 42°)(tan 16° tan 74°)
= (cot 42°/cot 42°)(cot 74°/cot 74°)
= 1 = RHS
Thus, tan 48° tan 16° tan 42° tan 74° = 1.
Question 7.
Show that
B. cos36°cos54° - sin36°sin54°
Answer:
We have
LHS = cos 36° cos 54° - sin 36° sin 54°
We know the trigonometric identity,
cos (90° - θ) = sin θ
First, replace θ by 54°.
We get, cos (90° - 54°) = sin 54°
⇒ cos 36° = sin 54° …(1)
Now, replace θ by 36°.
We get, cos (90° - 36°) = sin 36°
⇒ cos 54° = sin 36° …(2)
Using equations (1) & (2), we get
LHS = cos 36° cos 54° - sin 36° sin 54° = sin 54° sin 36° - sin 54° sin 36°
= 0 = RHS
Thus, cos 36° cos 54° - sin 36° sin 54° = 0.
Question 8.
If tan2A = cot(A – 18°) where 2A is an acute angle. Find the value of A.
Answer:
Given that, 2A is an acute angle.
⇒ 2A < 90°
So, using trigonometric identity, we can say that
cot (90° - 2A) = tan 2A [∵, cot (90° - θ) = tan θ]
Now, replace tan 2A by cot (90° - 2A) in the given question.
tan 2A = cot (A – 18°)
⇒ cot (90° - 2A) = cot (A – 18°)
Now, we can compare the degrees from above, we get
90° - 2A = A – 18°
⇒ 2A + A = 90° + 18°
⇒ 3A = 108°
⇒ A = 108°/3
⇒ A = 36°
Thus, the value of A is 36°.
Question 9.
If tan A = cot B where A and B are acute angles, prove that A + B = 90°
Answer:
Given that, A and B are acute angles.
⇒ A < 90° & B < 90°
So, using trigonometric identity, we can say
tan (90° - B) = cot B [∵, tan (90° - θ) = cot θ]
Replace cot B of RHS by tan (90° - B) in the given question.
tan A = cot B
⇒ tan A = tan (90° - B)
Now, comparing the degrees from the above, we get
A = 90° - B
⇒ A + B = 90°
Hence, proved that A + B = 90°.
Question 10.
If A, B and C are interior angles of a triangle ABC, then show that
Answer:
If A, B and C are interior angles of ∆ABC, then we can say that
A + B + C = 180° (by angle sum property of a triangle)
⇒ A + B = 180° - C
Take LHS:
⇒
⇒
[∵, tan (90° - θ) = cot θ, where θ = C/2 here]
Hence, .
Question 11.
Express sin 75° + cos 65° in terms of trigonometric ratios of angles between 0o and 45o.
Answer:
Given: sin 75° + cos 65°.
We can write,
75° = 90° - 15°
& 65° = 90° - 25°
Then, sin 75° + cos 65° = sin (90° - 15°) + cos (90° - 25°)
⇒ sin 75° + cos 65° = cos 15° + sin 25°
[∵, sin (90° - θ ) = cos θ & cos (90° - θ) = sin θ]
In cos 15° + sin 25°, 15° & 25° both are angles between 0° and 45°.
Thus, answer is sin 75° + cos 65° = cos 15° + sin 25°.
Exercise 11.4
Question 1.Evaluate the following :
A. (1 + tanθ + secθ)(1 + cotθ – cosecθ)
Answer:We have
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒
[∵, (a + b)(a – b) = a2 – b2]
⇒
[∵, (a + b)2 = a2 + b2 + 2ab]
⇒
[∵, sin2 θ + cos2 θ = 1]
⇒ ![](data:image/png;base64,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)
Thus, (1 + tan θ + sec θ)(1 + cot θ – cosec θ) = 2.
Question 2.Evaluate the following :
B. (sinθ + cosθ)2 + (sinθ - cosθ)2
Answer:We have
(sin θ + cos θ)2 + (sin θ – cos θ)2 = ((sin θ + cos θ) + (sin θ – cos θ))2 – 2(sin θ + cos θ)(sin θ – cos θ) [∵, a2 + b2 = (a + b)2 – 2ab]
⇒ (sin θ + cos θ)2 + (sin θ – cos θ)2 = (sin θ + cos θ + sin θ – cos θ)2 – 2(sin2 θ – cos2 θ)
[∵, (a + b)(a – b) = a2 – b2]
⇒ (sin θ + cos θ)2 + (sin θ – cos θ)2 = (2 sin θ)2 – 2 sin2 θ + 2 cos2 θ
⇒ (sin θ + cos θ)2 + (sin θ – cos θ)2 = 4 sin2 θ – 2 sin2 θ + 2 cos2 θ
⇒ (sin θ + cos θ)2 + (sin θ – cos θ)2 = 2 sin2 θ + 2 cos2θ
⇒ (sin θ + cos θ)2 + (sin θ – cos θ)2 = 2 (sin2 θ + cos2 θ)
⇒ (sin θ + cos θ)2 + (sin θ – cos θ)2 = 2 [∵, sin2 θ + cos2 θ = 1]
Thus, (sin θ + cos θ)2 + (sin θ – cos θ)2 = 2.
Question 3.Evaluate the following :
C. (sec2θ – 1) (cosec2θ – 1)
Answer:We have
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒
[∵, (1 – cos2 θ) = sin2 θ & (1 – sin2 θ = cos2 θ]
⇒ (sec2 θ – 1)(cosec2 θ – 1) = 1
Thus, (sec2 θ – 1)(cosec2 θ – 1) = 1.
Question 4.Show that ![](data:image/png;base64,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)
Answer:Using trigonometric identities,
cosec θ = 1/sin θ & cot θ = cos θ/sin θ
LHS = (cosec θ – cot θ)2
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
As we know, sin2 θ = 1 – cos2 θ
And (1 – cos2 θ) = (1 + cos θ)(1 – cos θ) [by (a2 – b2) = (a + b)(a – b)]
⇒ sin2 θ = (1 + cos θ)(1 – cos θ) …(i)
So, using equation (i),
LHS = (cosec θ – cot θ)2 = ![](data:image/png;base64,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)
= RHS
Hence, we have got
(cosec θ – cot θ)2 =
.
Question 5.Show that ![](data:image/png;base64,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)
Answer:Using trigonometric identity,
sin2 A + cos2 A = 1
⇒ cos2 A = 1 – sin2 A
Take Left hand side:
LHS = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= sec A + tan A = RHS
Thus,
.
Question 6.Show that ![](data:image/png;base64,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)
Answer:Using trigonometric identity, we have
cot A = 1/tan A
Take left hand side,
LHS = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGkAAAA/CAMAAADUixbiAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjqQOmaQOma2OpDbZgAAZgA6ZjoAZjo6ZjpmZjqQZma2ZpCQZpDbZrbbZrb/kDoAkDo6kGY6kGaQkJA6kJBmkLaQkLbbkNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///bpVih+wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACD0lEQVRYR+1Y23aCMBBMsK20Vq29qdTeREVaEsn//1w3CQIFWRq59HgkT4ghM7O7CcMSctLje0LpsA0F4d0n8a1lG1CAwa+NkL7ujaanRXij+JdwrgJc325CextkinCS1bLT/FSaPFqCxG/XHork03kRDy8FFD68lWkiBEMSDoXRByzfppcfg0CsgLtvWwDPAIjtaazmOGHFFp/iaqqeNSMMILdL1nufyaSEY0kiQuKDoCqScKQgwm3IFrfluh6FS1fdTcrhZQlcSgsrRcaVNNXYPxaOVUG4Mpe6puVltkwYTGJGSPnc6+e1MkVJgWh18dCBrIikBStlGk6BsN/lqrYVKy7SPSU0lbogFJJKkA5SVCbRCvxGssnozIdHlhWSSsjK7nEDcgL/eTziTzqEbl+8xts9KpqtIlI8xMKGEF9Mi2Zw2EZrUGJbU+LSYRBlTN3UQyyo2l1/SRRKpfuzi8DJRCB/drZKPT6567tok/8/R6+aVHMPe6Tfy3jYZv1e2sM26vfg1Ra/khr1e+ABE2vZpN8DRQlQs34v5WFFo34v7WE7v3f4wOj8XrWDtHv6rCJQn1VRK51V7MzEmjsUs/Xj2cYOpQwH60gZOZQyIIJ0pIwdCvqBjXSkzB0KLquwI3WEQ8G7BsUdKXOHgiIVd6SOcCgoUq0dKRSp1o4UilSrQ8GRautIle3criOVj9AP0Q9YOMdNjnMAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
= tan2 A = RHS
Thus,
= tan2 A.
Question 7.Show that ![](data:image/png;base64,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)
Answer:Take left hand side of the given question:
LHS = 1/cos θ – cos θ
= (1 – cos2 θ)/cos θ
= sin2 θ/cos θ [∵, sin2 θ + cos2 θ = 1 ⇒ sin2 θ = 1 – cos2 θ]
= sin θ × sin θ/cos θ
= sin θ × tan θ [∵, tan θ = sin θ/cos θ]
= tan θ sin θ = RHS
Thus,
.
Question 8.Simplify secA(1 – sinA)(secA + tanA)
Answer:By trigonometric identities, sec A = 1/cos A & tan A = sin A/cos A
Using these identities, we have
sec A (1 – sin A)(sec A + tan A) = ![](data:image/png;base64,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)
⇒ sec A (1 – sin A)(sec A + tan A) = ![](data:image/png;base64,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)
⇒ sec A (1 – sin A)(sec A + tan A) = ![](data:image/png;base64,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)
⇒ sec A (1 – sin A)(sec A + tan A) = ![](data:image/png;base64,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)
⇒ sec A (1 – sin A)(sec A + tan A) =
[∵, sin2 A + cos2 A = 1 ⇒ cos2 A = 1 – sin2 A]
⇒ sec A (1 – sin A)(sec A + tan A) = 1
Thus, sec A (1 – sin A)(sec A + tan A) = 1.
Question 9.Prove that (sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A
Answer:Take left hand side of the given equation:
LHS = (sin A + cosec A)2 + (cos A + sec A)2
Expanding the squares by formula: (a + b)2 = a2 + b2 + 2ab
= sin2 A + cosec2 A + 2 sin A cosec A + cos2 A + sec2 A + 2 cos A sec A
Rearranging the terms, we get,
= (sin2 A + cos2 A) + 2 sin A cosec A + 2 cos A sec A + cosec2 A + sec2 A
we know that,
![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= 5 + 1/(sin2 A cos2 A) …(i)
Now, take right hand side of the equation:
RHS = 7 + tan2 A + cot2 A
![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= 5 + 1/(sin2 A cos2 A) …(ii)
From equation (i) & (ii),
LHS = RHS
Hence, proved.
Question 10.Simplify (1 – cosθ) (1 + cosθ) (1 + cot2θ)
Answer:We have
(1 – cos θ)(1 + cos θ)(1 + cot2 θ) = [(1 – cos θ)(1 + cos θ)](1 + cot2 θ)
⇒ (1 – cos θ)(1 + cos θ)(1 + cot2 θ) = (1 – cos2 θ)(1 + cot2 θ) [∵, (a + b)(a – b) = a2 – b2]
⇒ (1 – cos θ)(1 + cos θ)(1 + cot2 θ) = sin2 θ × (1 + cos2 θ/sin2 θ) [∵, (1 – cos2 θ) = sin2 θ & cot2 θ = cos2 θ/sin2 θ]
⇒ (1 – cos θ)(1 + cos θ)(1 + cot2 θ) = sin2 θ × (sin2 θ + cos2 θ)/sin2 θ
⇒ (1 – cos θ)(1 + cos θ)(1 + cot2 θ) = sin2 θ + cos2 θ
⇒ (1 – cos θ)(1 + cos θ)(1 + cot2 θ) = 1 [∵, sin2 θ + cos2 θ = 1]
Thus, (1 – cos θ)(1 + cos θ)(1 + cot2 θ) = 1.
Question 11.If secθ + tanθ = p, then what is the value of secθ – tanθ ?
Answer:Given that, sec θ + tan θ = p.
By trigonometric identity, we have
sec2 θ – tan2 θ = 1
So, sec2 θ – tan2 θ = 1
⇒ (sec θ – tan θ) (sec θ + tan θ) = 1
⇒ sec θ – tan θ = 1/(sec θ + tan θ)
⇒ sec θ – tan θ = 1/p [given]
Hence, sec θ – tan θ = 1/p.
Question 12.If cosesθ + cotθ = k then prove that ![](data:image/png;base64,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)
Answer:Given that, cosec θ + cot θ = k
⇒
[∵, cosec θ = 1/sin θ & cot θ = cos θ/sin θ]
⇒ (1 + cos θ)/sin θ = k
⇒ 1 + cos θ = k sin θ
Squaring both sides, we get
(1 + cos θ)2 = (k sin θ)2
⇒ (1 + cos θ)2 = k2 sin2 θ
⇒ (1 + cos θ)2 = k2 (1 – cos2 θ) [∵, sin2 θ + cos2 θ = 1 ⇒ sin2 θ = 1 – cos2 θ]
⇒ (1 + cos θ)2 = k2 (1 – cos θ) (1 + cos θ) [∵, a2 – b2 = (a + b) (a – b)]
⇒ 1 + cos θ = k2 (1 – cos θ)
⇒ 1 + cos θ = k2 – k2 cos θ
⇒ k2 cos θ + cos θ = k2 – 1
⇒ cos θ (k2 + 1) = k2 – 1
⇒ cos θ = (k2 – 1)/(k2 + 1)
Thus, cos θ = (k2 – 1)/(k2 + 1).
Hence, proved.
Evaluate the following :
A. (1 + tanθ + secθ)(1 + cotθ – cosecθ)
Answer:
We have
⇒
⇒
⇒ [∵, (a + b)(a – b) = a2 – b2]
⇒ [∵, (a + b)2 = a2 + b2 + 2ab]
⇒ [∵, sin2 θ + cos2 θ = 1]
⇒
Thus, (1 + tan θ + sec θ)(1 + cot θ – cosec θ) = 2.
Question 2.
Evaluate the following :
B. (sinθ + cosθ)2 + (sinθ - cosθ)2
Answer:
We have
(sin θ + cos θ)2 + (sin θ – cos θ)2 = ((sin θ + cos θ) + (sin θ – cos θ))2 – 2(sin θ + cos θ)(sin θ – cos θ) [∵, a2 + b2 = (a + b)2 – 2ab]
⇒ (sin θ + cos θ)2 + (sin θ – cos θ)2 = (sin θ + cos θ + sin θ – cos θ)2 – 2(sin2 θ – cos2 θ)
[∵, (a + b)(a – b) = a2 – b2]
⇒ (sin θ + cos θ)2 + (sin θ – cos θ)2 = (2 sin θ)2 – 2 sin2 θ + 2 cos2 θ
⇒ (sin θ + cos θ)2 + (sin θ – cos θ)2 = 4 sin2 θ – 2 sin2 θ + 2 cos2 θ
⇒ (sin θ + cos θ)2 + (sin θ – cos θ)2 = 2 sin2 θ + 2 cos2θ
⇒ (sin θ + cos θ)2 + (sin θ – cos θ)2 = 2 (sin2 θ + cos2 θ)
⇒ (sin θ + cos θ)2 + (sin θ – cos θ)2 = 2 [∵, sin2 θ + cos2 θ = 1]
Thus, (sin θ + cos θ)2 + (sin θ – cos θ)2 = 2.
Question 3.
Evaluate the following :
C. (sec2θ – 1) (cosec2θ – 1)
Answer:
We have
⇒
⇒ [∵, (1 – cos2 θ) = sin2 θ & (1 – sin2 θ = cos2 θ]
⇒ (sec2 θ – 1)(cosec2 θ – 1) = 1
Thus, (sec2 θ – 1)(cosec2 θ – 1) = 1.
Question 4.
Show that
Answer:
Using trigonometric identities,
cosec θ = 1/sin θ & cot θ = cos θ/sin θ
LHS = (cosec θ – cot θ)2
As we know, sin2 θ = 1 – cos2 θ
And (1 – cos2 θ) = (1 + cos θ)(1 – cos θ) [by (a2 – b2) = (a + b)(a – b)]
⇒ sin2 θ = (1 + cos θ)(1 – cos θ) …(i)
So, using equation (i),
LHS = (cosec θ – cot θ)2 =
= RHS
Hence, we have got
(cosec θ – cot θ)2 = .
Question 5.
Show that
Answer:
Using trigonometric identity,
sin2 A + cos2 A = 1
⇒ cos2 A = 1 – sin2 A
Take Left hand side:
LHS =
= sec A + tan A = RHS
Thus, .
Question 6.
Show that
Answer:
Using trigonometric identity, we have
cot A = 1/tan A
Take left hand side,
LHS =
= tan2 A = RHS
Thus, = tan2 A.
Question 7.
Show that
Answer:
Take left hand side of the given question:
LHS = 1/cos θ – cos θ
= (1 – cos2 θ)/cos θ
= sin2 θ/cos θ [∵, sin2 θ + cos2 θ = 1 ⇒ sin2 θ = 1 – cos2 θ]
= sin θ × sin θ/cos θ
= sin θ × tan θ [∵, tan θ = sin θ/cos θ]
= tan θ sin θ = RHS
Thus, .
Question 8.
Simplify secA(1 – sinA)(secA + tanA)
Answer:
By trigonometric identities, sec A = 1/cos A & tan A = sin A/cos A
Using these identities, we have
sec A (1 – sin A)(sec A + tan A) =
⇒ sec A (1 – sin A)(sec A + tan A) =
⇒ sec A (1 – sin A)(sec A + tan A) =
⇒ sec A (1 – sin A)(sec A + tan A) =
⇒ sec A (1 – sin A)(sec A + tan A) = [∵, sin2 A + cos2 A = 1 ⇒ cos2 A = 1 – sin2 A]
⇒ sec A (1 – sin A)(sec A + tan A) = 1
Thus, sec A (1 – sin A)(sec A + tan A) = 1.
Question 9.
Prove that (sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A
Answer:
Take left hand side of the given equation:
LHS = (sin A + cosec A)2 + (cos A + sec A)2
Expanding the squares by formula: (a + b)2 = a2 + b2 + 2ab= sin2 A + cosec2 A + 2 sin A cosec A + cos2 A + sec2 A + 2 cos A sec A
Rearranging the terms, we get,= (sin2 A + cos2 A) + 2 sin A cosec A + 2 cos A sec A + cosec2 A + sec2 A
we know that,
=
=
= 5 + 1/(sin2 A cos2 A) …(i)
Now, take right hand side of the equation:
RHS = 7 + tan2 A + cot2 A
=
=
=
=
= 5 + 1/(sin2 A cos2 A) …(ii)
From equation (i) & (ii),
LHS = RHS
Hence, proved.
Question 10.
Simplify (1 – cosθ) (1 + cosθ) (1 + cot2θ)
Answer:
We have
(1 – cos θ)(1 + cos θ)(1 + cot2 θ) = [(1 – cos θ)(1 + cos θ)](1 + cot2 θ)
⇒ (1 – cos θ)(1 + cos θ)(1 + cot2 θ) = (1 – cos2 θ)(1 + cot2 θ) [∵, (a + b)(a – b) = a2 – b2]
⇒ (1 – cos θ)(1 + cos θ)(1 + cot2 θ) = sin2 θ × (1 + cos2 θ/sin2 θ) [∵, (1 – cos2 θ) = sin2 θ & cot2 θ = cos2 θ/sin2 θ]
⇒ (1 – cos θ)(1 + cos θ)(1 + cot2 θ) = sin2 θ × (sin2 θ + cos2 θ)/sin2 θ
⇒ (1 – cos θ)(1 + cos θ)(1 + cot2 θ) = sin2 θ + cos2 θ
⇒ (1 – cos θ)(1 + cos θ)(1 + cot2 θ) = 1 [∵, sin2 θ + cos2 θ = 1]
Thus, (1 – cos θ)(1 + cos θ)(1 + cot2 θ) = 1.
Question 11.
If secθ + tanθ = p, then what is the value of secθ – tanθ ?
Answer:
Given that, sec θ + tan θ = p.
By trigonometric identity, we have
sec2 θ – tan2 θ = 1
So, sec2 θ – tan2 θ = 1
⇒ (sec θ – tan θ) (sec θ + tan θ) = 1
⇒ sec θ – tan θ = 1/(sec θ + tan θ)
⇒ sec θ – tan θ = 1/p [given]
Hence, sec θ – tan θ = 1/p.
Question 12.
If cosesθ + cotθ = k then prove that
Answer:
Given that, cosec θ + cot θ = k
⇒ [∵, cosec θ = 1/sin θ & cot θ = cos θ/sin θ]
⇒ (1 + cos θ)/sin θ = k
⇒ 1 + cos θ = k sin θ
Squaring both sides, we get
(1 + cos θ)2 = (k sin θ)2
⇒ (1 + cos θ)2 = k2 sin2 θ
⇒ (1 + cos θ)2 = k2 (1 – cos2 θ) [∵, sin2 θ + cos2 θ = 1 ⇒ sin2 θ = 1 – cos2 θ]
⇒ (1 + cos θ)2 = k2 (1 – cos θ) (1 + cos θ) [∵, a2 – b2 = (a + b) (a – b)]
⇒ 1 + cos θ = k2 (1 – cos θ)
⇒ 1 + cos θ = k2 – k2 cos θ
⇒ k2 cos θ + cos θ = k2 – 1
⇒ cos θ (k2 + 1) = k2 – 1
⇒ cos θ = (k2 – 1)/(k2 + 1)
Thus, cos θ = (k2 – 1)/(k2 + 1).
Hence, proved.