Question

If cosecθ + cotθ = 5, then evaluate secθ.

Answer 1

cosecθ + cotθ = 5

∴ `1/sintheta + cos theta/sin theta` = 5

∴ `(1+costheta)/sintheta=5`

∴ 1 + cosθ = 5sinθ

Squaring both the sides, we get

∴ 1 + 2cosθ + cosθ² = 25sin²θ

∴ 1 + 2cosθ + cos²θ = 25(1 – cos²θ)

∴ 1 + 2cosθ + cos²θ = 25 – 25cos²θ

∴ 26cos²θ + 2cosθ – 24 = 0

∴ 26cos²θ + 26cosθ − 24cosθ – 24 = 0

∴ 26cosθ (cosθ  + 1) – 24(cosθ + 1) = 0

∴ (cosθ + 1) (26cosθ – 24) = 0

∴ cosθ + 1 = 0 or 26cosθ – 24 = 0

∴ cosθ = – 1  or cosθ  = `24/26 = 12/13`

When cosθ = – 1, sinθ = 0

∴ cosecθ and cotθ are not defined.

∴ cosθ ≠ – 1

Hence, cosθ = `12/13`

∴ secθ = `13/12`