Prove the identity: $\left(\sin \alpha - \sin \beta\right) \left(\sin \alpha + \sin \beta\right) = \sin \left(\alpha - \beta\right) \sin \left(\alpha + \beta\right)$
$\begin{aligned}
RHS & = \sin \left(\alpha - \beta\right) \sin \left(\alpha + \beta\right) \\\\
& = \left(\sin \alpha \cos \beta - \cos \alpha \sin \beta\right) \left(\sin \alpha \cos \beta + \cos \alpha \sin \beta\right) \\\\
& = \sin^2 \alpha \cos^2 \beta - \cos^2 \alpha \sin^2 \beta \\\\
& = \sin^2 \alpha \left(1 - \sin^2 \beta\right) - \sin^2 \beta \left(1 - \sin^2 \alpha\right) \\\\
& = \sin^2 \alpha - \sin^2 \alpha \sin^2 \beta - \sin^2 \beta + \sin^2 \alpha \sin^2 \beta \\\\
& = \sin^2 \alpha - \sin^2 \beta \\\\
& = \left(\sin \alpha - \sin \beta\right) \left(\sin \alpha + \sin \beta\right) = LHS
\end{aligned}$
Hence proved.