If you have more than one question you need to make separate posts.
In order to prove this identity, you need to start with one side, and then using existing identities, end up at the other side of the identity.
`LS=cos^2x-sin^2x` use `sin^2x+cos^2x=1`
`=cos^2x-(1-cos^2x)` remove brackets
`=cos^2x-1+cos^2x` collect like terms
The identity has been proven.