Your sum is Re(S).
First of all I'm not sure if this is the right place to post this question. It's for a partial differential equations course, but it doesn't seem to have anything to do with differential equations directly.
I need to prove that 1 + 2 * Sum(n=1,N)(cos(nx)) = sin((N + (1/2))x)/sin((1/2)x).
I have no idea where to start, other than that we are studying Fourier series. Can someone please help me figure out how to get started?