trig help. How do you prove identities?

Question

I've been staring at this problem for an hour and going over my notes but I can't figure out how to start it off. Can someone help me out please and maybe give me a first step?

problem:

prove the identity

cos²x-sin²x=1-2sin²x

right now I'm trying to prove that they equal each other. So since right now you don't know for a fact they are equal you can't take things and put them on the other side of the equal side. you just have to get one side to equal the other. for instance

cos²x+sin²x +1/tanx= 1 + cotx
for this one you take the identity and cos²x+sin²x=1 so you have 1+1/tanx+1+cotx then you use the reciprocal identity and 1/tanx=cotx so you get 1 + cotx=1 + cotx

Answer 1

add 2sin²x to both sides, and you get cos²x + sin²x = 1, which is of course an identity.

Answer 2

cos^2(x) - sin^2(x) = 1 - 2sin^2(x)

The normaly way to prove identities is to choose the most complex side of the equation, and work with that to try and get the other side. In this case, it's a toss-up, but we'll choose the left hand side.

LHS = cos^2(x) - sin^2(x)

Note that we're TRYING to make it look like the right hand side. Let's use the identity cos^2(y) = 1 - sin^2(y). Using this (instead, with x)

LHS = 1 - sin^2(x) - sin^2(x)

We can combine those two sines just as if we were combining
-z - z, which is equal to -2z.

LHS = 1 - 2sin^2(x) = RHS