$\cos (A+B)+\cos (A-B)$ is equal to:

Using standard trigonometric compound angle expansion formulas:
$$\cos (A+B)=\cos A\cos B-\sin A\sin B$$
$$\cos (A-B)=\cos A\cos B+\sin A\sin B$$
Adding these two identities together:
$$\cos (A+B)+\cos (A-B)=(\cos A\cos B-\sin A\sin B)+(\cos A\cos B+\sin A\sin B)$$
The terms $-\sin A\sin B$ and $+\sin A\sin B$ cancel each other out:
$$\cos (A+B)+\cos (A-B)=2\cos A\cos B$$