There was something that I was getting a little curious about. We know that there are the so-called compound-angle formulae for calculating sines and cosines of sums of angles in terms of those of the original angles:
$$\sin (\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$$ $$\cos (\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta.$$
Now, do there exist any sort of formulae for products of angles, that is, for $\sin(\alpha\beta)$ and $\cos(\alpha\beta)$? If so, how can one derive them? (I had a play and tried to get something, and it ended up being rather nasty; only holding for small-ish angles $\beta$ and resulting in some infinite series that may or may not converge to anything well-known.)