TITLE: Prove $\frac{1}{2} + \cos(x) + \cos(2x) + \dots+ \cos(nx) = \frac{\sin(n+\frac{1}{2})x}{2\sin(\frac{1}{2}x)}$ for $x \neq 0, \pm 2\pi, \pm 4\pi,\dots$
QUESTION [5 upvotes]: I know that this can be proven inductively. However, I can't get passed the trig. I am pretty sure trig identities can show that the expression above is true for $n=0$, and that if the expression holds for $n=k$ it holds for $n=k+1$. But alas, I am getting lost in a sea of trig. Hopefully someone can shed some light on this.

REPLY [6 votes]: Hint:
$$\frac{1}{2} + \sum_{k=1}^n \cos(kx)
= \frac{1}{2}\sum_{k=-n}^n e^{ikx}$$
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