It's always a pain (to me anyway) to remember the formulae for the expansions of and its ilk.

The usual geometric derivation in trig books is not one that leaps merrily forth from my pencil, either.

Some time ago I came across this nice little derivation using DeMoivre's Theorem. It's the sort of thing that you can scribble out fast on scratch paper just to make sure that you're getting the signs right.

Start with:

Straightforward enough, right? Now expand using :

Do the algebra:

And separate the real and imaginary parts:

There are two of your favorite trig identities. Try the same thing with:

You get the idea.