Question

How do you verify `(1+cotalpha)^2-2cotalpha=1/((1-cosalpha)(1+cosalpha))`?

Answer 1

Start by transforming all the terms into sine and cosine using the identity `color(red)(cot theta = 1/tantheta = 1/(sintheta/costheta) = costheta/sintheta`

`(1 + cosalpha/sinalpha)^2 - (2cosalpha)/sinalpha = 1/((1 - cosalpha)(1 + cosalpha))`

`1 + (2cosalpha)/sinalpha + cos^2alpha/sin^2alpha - (2cosalpha)/sinalpha = 1/(1- cos^2alpha)`

The `(2cosalpha)/sinalpha`'s cancel each other out

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

We use the identity `color(red)(sin^2beta + cos^2beta =1-> sin^2beta = 1- cos^2beta` at this point in the process.

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

`1/sin^2alpha = 1/sin^2alpha`

`LHS = RHS`

Identity Proved!

Hopefully this helps!

Answer 2

`LHS=(1+cotalpha)^2-2cotalpha`

`=1^2+cot^2alpha+2cotalpha-2cotalpha`

`=csc^2alpha`

`=1/sin^2alpha`

`=1/(1-cos^2alpha)`

`=1/((1-cosalpha)(1+cosalpha))=RHS`

Proved