Building high frequencies out of low frequencies

If you have sines and cosines of some fundamental frequency, and you're able to take products and sums, then you can construct sines and cosines at any multiple of the fundamental frequency.

Here's a proof.

Taking real parts gives us cos n in the first equation and the even terms of the sum in the last equation.

Taking imaginary parts gives us sin n in the first equation and the odd terms of the sum in the last equation.