Question

Prove the following trigonometric identities.

`sqrt((1 - cos theta)/(1 + cos theta)) = cosec theta - cot theta`

Answer 1

We know that, `sin^2 theta + cos^2 theta = 1`

Multiplying numerator and denominator under the square root by `1 - cos theta)` we have

`sqrt((1 - cos theta)/(1 + cos theta)) = sqrt(((1 - cos theta)(1 - cos theta))/((1 + cos theta)(1 - cos theta)))`

`= sqrt((1 - cos theta)^2/(1 - cos^2 theta))`

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

`= (1 - cos theta)/sin theta`

`= 1/sin theta - cos theta/sin theta`

`= cosec theta - cot theta`