Question

Prove the following trigonometric identities.

`((1 + cot^2 theta) tan theta)/sec^2 theta = cot theta`

Answer 1

We have to prove `((1 + cot^2 theta) tan theta)/sec^2 theta = cot theta`

We know that, `sec^2 theta - tan^2 theta = 1`

So

`((1 + cot^2 theta)tan theta)/sec^2 theta = ((1 + cot^2 theta)tan theta)/(1 + tan^2 theta)`

` = ((1 + 1/tan^2 theta)tan theta)/(1 + tan^2 theta)`

`= (((tan^2 theta + 1)/(tan^2 theta)) tan theta)/(1 + tan^ 2 theta)`

`= ((1 + tan^2 theta)tan theta)/(tan^2 theta(1 + tan^2 theta))`

`= 1/tan theta`

`= cot theta`