Prove that cosycos(x-y)-sinysin(x-y)=cosx?

Recall the identity $\cos \left(A + B\right) = \cos A \cos B - \sin A \sin B$
Putting $A = y$ and $B = x - y$, we get $A + B = y + x - y = x$ and the above identity becomes
$\cos y \cos \left(x - y\right) - \sin y \sin \left(x - y\right) = \cos x$