How do you prove ##cos2x=cos^2x-sin^2## using other trigonometric identities?

Apply the angle-sum identity for cosine to ##cos(x+x)##.

The identity needed is the angle-sum identity for cosine.

##cos(alpha + beta) = cos(alpha)cos(beta) - sin(alpha)sin(beta)##

With that, we have

##cos(2x) = cos(x + x)##

##= cos(x)cos(x) - sin(x)sin(x)##

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