Answer
See the full explanation below.
Work Step by Step
Let us consider the left side of the given expression:
$\cos \left( \alpha +\beta \right)+\cos \left( \alpha -\beta \right)$
By using the identities of trigonometry,
$\cos \,\left( \alpha +\beta \right)=\cos \,\alpha \,\cos \beta -\sin \,\alpha \,\sin \,\beta $
$\cos \,\left( \alpha -\beta \right)=\cos \,\alpha \,\cos \beta +\sin \,\alpha \,\sin \,\beta $ , the above expression can be further simplified as:
$\begin{align}
& \cos \,\left( \alpha +\beta \right)+\cos \,\left( \alpha -\beta \right)=\left( \cos \,\alpha \,\cos \beta -\sin \,\alpha \,\sin \,\beta \right)+\left( \cos \,\alpha \,\cos \beta +\sin \,\alpha \,\sin \,\beta \right) \\
& =\cos \,\alpha \,\cos \beta +\cos \,\alpha \,\cos \beta -\sin \,\alpha \,\sin \,\beta +\sin \,\alpha \,\sin \,\beta \\
& =2\cos \,\alpha \,\cos \beta
\end{align}$
Hence, the left side of the given expression is equal to the right side, which is $\cos \left( \alpha +\beta \right)+\cos \left( \alpha -\beta \right)=2\cos \,\alpha \cos \,\beta $.
