Answer
$\cos \left( x+y\right) \cos \left( x-y\right) =\left( \cos x\cos y-\sin x\sin y\right) \left( \cos x\cos y+\sin x\sin y\right)=\cos ^{2}x\cos ^{2}y-\sin ^{2}x\sin ^{2}y=\cos ^{2}x\left( 1-\sin ^{2}y\right) -\sin ^{2}x\sin ^{2}y =\cos ^{2}x-\sin ^{2}y\left( \sin ^{2}x+\cos ^{2}x\right) =\cos ^{2}x-\sin ^{2}y $
Work Step by Step
$\cos \left( x+y\right) \cos \left( x-y\right) =\left( \cos x\cos y-\sin x\sin y\right) \left( \cos x\cos y+\sin x\sin y\right)=\cos ^{2}x\cos ^{2}y-\sin ^{2}x\sin ^{2}y=\cos ^{2}x\left( 1-\sin ^{2}y\right) -\sin ^{2}x\sin ^{2}y =\cos ^{2}x-\sin ^{2}y\left( \sin ^{2}x+\cos ^{2}x\right) =\cos ^{2}x-\sin ^{2}y $
