Question

Prove the following trigonometric identities.

(1 + tan²θ) (1 − sinθ) (1 + sinθ) = 1

Answer 1

We have to prove  `(1 + tan^2 theta)(1 - sin theta)(1 + sin theta) = 1`

We know that

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

`sec^2 theta - tan^2 theta = 1`

So

`(1 + tan^2 theta)(1 - sin theta) = (1 + tan^2 theta){(1 - sin theta)(1 + sin theta)}`

` = (1 + tan^2 theta)(1 - sin^2 theta)`

`= sec^2 theta cos^2 theta`

` = 1/cos^2 theta cos^2 theta`

= 1