Answer
Please refer to the step-by-step part below.
Work Step by Step
Recall:
(1) $\sin{(\alpha+\beta)}=\sin{(\alpha)}\cos{(\beta)}+\cos{(\alpha)}\sin{(\beta)}$
(2) $\sin{(\alpha-\beta)}=\sin{(\alpha)}\cos{(\beta)}-\cos{(\alpha)}\sin{(\beta)}$
Hence,
$\sin{(\alpha+\beta)}+\sin{(\alpha-\beta)}\\
=[\sin{(\alpha)}\cos{(\beta)}+\cos{(\alpha)}\sin{(\beta)}]+[\sin{(\alpha)}\cos{(\beta)}-\cos{(\alpha)}\sin{(\beta)}]\\
=[\sin{(\alpha)}\cos{(\beta)}+\sin{(\alpha)}\cos{(\beta)}+\cos{(\alpha)}]+[\sin{(\beta)}-\cos{(\alpha)}\sin{(\beta)}]\\
=2\sin{(\alpha)}\cos{(\beta)} +0\\
=2\sin{(\alpha)}\cos{(\beta)}$