How do you prove sin²x - sin²y = sin(x + y)sin(x - y)?

We have to prove \(\sin (x+y)\sin (x-y)=sin^{2}x-sin^{2}y\)

Solution

Let us start with RHS

=\((sin x cos y+ cos x sin y)(sin x cos y – cos x sin y)\)

=\((\sin ^{2}x\cos ^{2}y-\cos ^{2}x\sin ^{2}y)\)

=\(\sin ^{2}x(1-\sin ^{2}y)-(1-\sin ^{2}x)\sin ^{2}y\)

=\(\sin ^{2}-\sin ^{2}y\)

= LHS

Hence Proved