Question

Prove the identity `sin theta/ (1-cos theta) - 1/sin theta -= 1/tan theta`?

Answer 1

We seek to prove that:

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

Consider the LHS:

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

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

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

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

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

`\ \ \ \ \ \ \ \ = ( cos theta) / (sin theta)`

`\ \ \ \ \ \ \ \ = cot theta`

`\ \ \ \ \ \ \ \ = 1/tan theta`

`\ \ \ \ \ \ \ \ = LHS \ \ \ \ QED`