taylor expand sin & cos and you're done. I think that's pretty much the most popular method.
Edit: this works b/c the series for sin & cos are uniformly convergent...not too difficult to prove.
You might also benefit from this: sin(z)=[exp(iz)-exp(-iz)]/2i, cos(z)=[exp(iz)+exp(-iz)]/2.
Another way I've seen is to define a complex # z=cos(t)+i*sin(t)
dz/dt = -sin(t) + i*cos(t)...now remember i^2 = -1, so we have i(i*sin(t) + cos(t)) = i*z
Now integrate. The integrating constant is clearly 0...then exponentiate and you're done.