Question

Prove the following trigonometric identities.

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

Answer 1

In the given question, we need to prove `(1 + cos theta - sin^2 theta)/(sin theta (1 + cos theta)) = cot theta`

Using the property  `sin^2 theta + cot^2 theta = 1` we get

So

`(1 + cos theta - sin^2 theta)/(sin theta (1 +  cos theta))`

`= (1 + cos theta - (1 - cos^2 theta))/(sin theta (1 + cos theta)`

`= (cos theta + cos^2 theta)/(sin theta (1 + cos theta))`

Solving further, we get

`(cos theta + cos^2 theta)/(sin(1 + cos theta)) = (cos theta (1 + cos theta))/(sin theta(1 + cos theta))`

`= cos theta/sin theta`

`= cot theta`

Hence proved.