Is this trigonometric identity too good to be true?

\sin(x y) \sin(x + y) = (\sin(x) \sin(y)) (\sin(x) + \sin(y)).

Let's see. We know that

\sin(x\pm y) = \sin(x)\cos(y)\pm \cos(x)\sin(y).

Multiplying the two gives

\begin{align} \sin(x y)\cdot\sin(x + y) &= \sin^{2} (x) \cdot\cos^{2}(y)-\cos^{2} (x) \cdot\sin^{2}(y) \\&=\sin^{2}(x)-\sin^{2}(x)\sin^{2}(y)-\sin^{2} (y)+\sin^{2}(y)\sin^{2}(x) \\&= \sin^{2}(x)-\sin^{2}(y) \\&= (\sin(x)-\sin(y))(\sin(x)+\sin(y))\end{align}