Answer
$\cos (x+y)+\cos(x-y)=2\cos x\cos y$
Work Step by Step
Start with the left side:
$\cos (x+y)+\cos(x-y)$
Expand using the addition and subtraction formulas for cosine:
$=(\cos x\cos y-\sin x\sin y)+(\cos x\cos y+\sin x\sin y)$
Simplify:
$=\cos x\cos y-\sin x\sin y+\cos x\cos y+\sin x\sin y$
$=\cos x\cos y+\cos x\cos y$
$=2\cos x\cos y$
Since this equals the right side, the identity has been proven.
