It depends where you want to start.
If you know the Taylor series for e^z, sin theta and cos theta, together with the basic properties of i = sqrt(-1), then you can easily find that:
e^(itheta) = cos theta + i sin theta
Then:
cos 2theta + isin 2theta = e^(2itheta) = (e^(itheta))^2
= (cos theta + isin theta)^2
= cos^2theta + 2i cos theta sin theta + i^2 sin^2 theta
= cos^2theta + 2i cos theta sin theta - sin^2 theta
= (cos^2theta - sin^2 theta) + i (2cos theta sin theta)
Comparing the real and imaginary parts we get two formulae for the price of one:
cos 2theta = cos^2 theta - sin^2 theta
sin 2theta = 2cos theta sin theta
