Proving $\cos(A + B)\cos(A - B) = -(\sin A + \cos B)(\sin A - \cos B)$. Could you give me some guidance on proving the following trig identity?
$$\cos(A + B)\cos(A - B) = -(\sin A + \cos B)(\sin A - \cos B).$$

You have $\cos(A+B)=\cos A \cos B - \sin A \sin B$ and $\cos(A-B)=\cos A \cos B + \sin A \sin B$, these are standard formulas.
Multiply them together and simplify as needed.
Them multiply out the RHS, and see what you get.
I won't finish it for you, because the exercise is worth doing by hand.
