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