Question

how do you soulve identity sin(x+y)sin(x-y) = sin^2x - sin^2y?

Answer 1

Write \(\sin(x+y)=A+B\) and \(\sin(x-y)=A-B\) where \(A=\sin x\cos y\) and \(B=\cos x\sin y\). Then
\[\begin{align*}\Large
\sin(x+y)\sin(x-y)&=A^2-B^2\\
&=\sin^2x\cos^2y-\cos^2x\sin^2y\\
&=(\sin^2x\cos^2y+\sin^2x\sin^2y)-(\sin^2x\sin^2y+\cos^2x\sin^2y)\\
&=\sin^2x-\sin^2y.
\end{align*}\]

Answer 2

thank you!