Indentifying $\sin(mx) = 2\cos(x)\sin\left[(m-1)x\right] - \sin\left[(m-2)x\right]$ I encountered in a work of Joseph Fourier's the identity:
$$\sin(mx) = 2\cos(x)\sin\left[(m-1)x\right] - \sin\left[(m-2)x\right]$$
which holds for all real $m$ and $x$.
I had trouble, however, locating this in common lists of trigonometric identities.  Does this identity have a name, and where can I find it listed?  If it is not actually a common identity, how is it derived?

or just observe:
$$
\sin ((m-1)x \pm x) = \sin(m-1)x \cos x \pm \sin x \cos(m-1)x
$$
and add the two equations together
