Question

Prove the following trigonometric identities.

(cosecθ + sinθ) (cosecθ − sinθ) = cot² θ + cos²θ

Answer 1

We have to prove  (cosecθ + sinθ) (cosecθ − sinθ) = cot² θ + cos²θ

We know that

`sin^2 theta + cos^2 theta = 1`

`cosec^2 theta - cot^2 theta = 1`

So,

`(cosec theta + sin theta)(cosec theta - sin theta) = cosec^2 theta -  sin^2 theta`

`= (1 + cot^2 theta) - (1 - cos^2 theta)`

`= 1 + cot^2 theta  - 1 + cos^2 theta`

`= cot^2 theta + cos^2 theta`