Question

Prove the following trigonometric identities:

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

Answer 1

We have to prove `sin theta/(1 - cos theta) = cosec theta + cot theta`

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

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

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

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

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

`= 1/sin theta + cos theta/sin theta`

`= cosec theta  + cot theta`