Answer
see proof below.
Work Step by Step
Apply the sum and difference formulas:
$\cos(A+B)=\cos(A)\cos(B) -\sin(A)\sin(B)$
and
$\cos(A-B)=\cos(A)\cos(B) +\sin(A) \ \sin(B)$
In order to prove the given identity, we simplify the left hand side $\text{LHS}$ as follows:
$\text{LHS}=\sin (\alpha+\beta) \sin (\alpha - \beta) \\
=\sin^2 \alpha \cos^2 \beta -\cos^2 \alpha \sin^2 \beta \\=\sin^2 \alpha -\sin^2 \alpha \sin^2 \beta-\sin^2 \beta+\sin^2 \alpha \sin^2 \beta \\ =\sin^2 \alpha-\sin^2 \beta \\=\text{ RHS}$
