This seems to be a bit dead. If I knew anything about rings I'd offer my own answer, but as is I'll just post my own:
Find a general identity for sin^n x for n odd. By identity, I mean rewriting in terms of a sum of single powers of the sine and cosine harmonics (sin(nx), cos(nx).
This problem is more useful than interesting, I realize, but I'm just trying to restart the game.