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)##