Expanding this out gives:

Simplify by using pythagorean identity:
sin^2 + cos^2 = 1

Next use cosine difference identity:
cos(x-y) = cos(x)cos(y) +sin(x)sin(y)

Next use cosine double angle identity:
cos(2x) = 2cos^2 - 1

Finally distribute and Right side = Left side

Identity is proved