<s>[INST] Prove $\cos(A + B) \cos(A − B) = \cos^2A − \sin^2B$ I have a question regarding trigonometric identities. The question I am currently struggling to understand is:

Prove that $$\cos(A + B) \cos(A − B) = \cos^2A − \sin^2B$$

When approaching this problem I know that there is
$$\cos2A = \cos^2A - \sin^2A$$
but how would I apply this here, or is it completely wrong way of approaching it, or should I try
$$\cos(A+B)=\cos A \cos B - \sin A \sin B$$
Many Thanks.
 [/INST] \begin{align*}
\cos(x+y)\cdot\cos(x-y)&=
(\cos x\cos y-\sin x\sin y)(\cos x\cos y+\sin x\sin y)\\
&=\cos^2 x\cos^2 y-\sin^2 x\sin^2 y\\
&=\cos^2 x(1-\sin^2 y)-(1-\cos^2 x)\sin^2 y\\
&=\cos^2 x-\cos^2 x\sin^2 y-\sin^2 y+\cos^2 x\sin^2y\\
&=\cos^2 x-\sin^2y
\end{align*}
</s>