4. If tan θ = 1, then find the value of (sinθ + cos θ) ÷ (secθ + cosecθ ), where θ is an acute angle.

Sol. tan θ  = 1

But we know that, tanθ = y/x = 1/1

∴ y = 1 and x = 1

r = √(x²+y² )

∴ r =√((1)²+(1)² )

∴ r =√(1+1)

∴ r =√2

Now,

sinθ = y/r = 1/√2

cosθ = x/r = 1/√2

sec θ = r/x = √2

cosec θ = r/y = √2

∴ (sinθ  + cos θ) ÷  (secθ  + cosecθ ) = (1/√2 + 1√2) ÷ (√2 + √2)

∴  (sinθ  + cos θ) ÷  (secθ  + cosecθ ) =(1+1)/√2 ÷ 2√2

∴ (sinθ  + cos θ) ÷  (secθ  + cosecθ ) = 2/√2 / 2√2

∴ (sinθ  + cos θ) ÷  (secθ  + cosecθ ) = 2/4

∴ (sinθ  + cos θ) ÷  (secθ  + cosecθ ) = 1/2