Answer
$(\cos x+\cos y)^2+(\sin x-\sin y)^2=2+2\cos(x+y)$
Work Step by Step
Start with the left side:
$(\cos x+\cos y)^2+(\sin x-\sin y)^2$
Expand:
$=(\cos^2 x+2\cos x\cos y+\cos^2 y)+(\sin^2 x-2\sin x\sin y+\sin^2 y)$
Rearrange terms:
$=(\cos^2 x+\sin^2 x)+(\cos^2 y+\sin^2y)+(2\cos x\cos y-2\sin x\sin y)$
Use the identity $\sin^2 x+\cos^2 x=1$:
$=1+1+2(\cos x\cos y-\sin x\sin y)$
Use the identity $\cos (x+y)=\cos x\cos y-\sin x\sin y$:
$=2+2\cos(x+y)$
Since this equals the right side, the identity is proven.
