Question

If cosecθ + cotθ = 5, then evaluate secθ.

Answer 1

cosecθ + cotθ = 5

∴ ${\displaystyle \frac{1}{\sin \theta }+\frac{\cos \theta }{\sin \theta }}$ = 5

∴ ${\displaystyle \frac{1+\cos \theta }{\sin \theta }=5}$

∴ 1 + cosθ = 5sinθ

Squaring both the sides, we get,

∴ (1 + cosθ)² = 25sin²θ

∴ 1 + 2cosθ + cos²θ = 25sin²θ

∴ 1 + 2cosθ + cos²θ = 25(1 – cos²θ)

∴ 1 + 2cosθ + cos²θ = 25 – 25cos²θ

∴ 1 + 2cosθ + cos²θ + 25cos²θ – 25 = 0 

∴ 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 = ${\displaystyle \frac{24}{26}=\frac{12}{13}}$

When cosθ = – 1, sinθ = 0

∴ cosecθ and cotθ are not defined.

∴ cosθ ≠ – 1

∴ cosθ = ${\displaystyle \frac{12}{13}}$ 

∴ secθ = ${\displaystyle \frac{13}{12}}$