Question

Prove the identity `(1/sinx - 1/tanx)^2 -= (1-cosx)/(1+cosx).` ?

Prove the identity `(1/sinx - 1/tanx)^2 -= (1-cosx)/(1+cosx).`

Answer 1

I would start with the left hand side, by rewriting in terms of sine and cosine.

LHS:

`(1/sinx - 1/(sinx/cosx))^2`

`(1/sinx - cosx/sinx)^2`

`((1 - cosx)/sinx)^2`

`(1 -cosx)^2/sin^2x`

Recall that

`sin^2x +cos^2x = 1 -> sin^2x= 1- cos^2x`.

`(1 - cosx)^2/(1 - cos^2x)`

Now do a little factoring.

`((1 - cosx)(1 - cosx))/((1 + cosx)(1 - cosx))`

`(1 - cosx)/(1 + cosx)`

We now see that LHS = RHS, therefore we've proven this identity.

Hopefully this helps!