Prove the identity: (cosx + cosy) ^2 + (sinx - siny) ^2 = 2 + 2 cos (x+y).

We expand first (cos x + cos y) ^2 + (sin x-sin y) ^2:

= cos^2 x + 2 cos x cos y + cos^2 y + sin ^2 x + sin ^2 y - 2 sin x sin y

= (cos^2 x + sin ^2 x) + (cos^2 y + sin ^2 y) + 2 (cos x cos y - sin x sin y)

then, apply the trigonometric identities of addition and summation of angles

= 1 + 1 + 2 cos (x+y)

we add the following identities above that results to

2 + 2 cos (x+y)