# 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 $x \neq 0, \pm 2\pi, \pm 4\pi,\dots$

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.

• Hint : $\sum_{k=1}^n cos(nx)=Re(\sum_{k=1}^n e^{inx})$ – Bertrand R Oct 9 '13 at 22:25
• Looks like you can derive this result using Fourier series – peterwhy Oct 9 '13 at 22:26
• @peterwhy That would be odd, since this is an important inequality to derive some identities concerning the Dirichlet kernel. – Pedro Tamaroff Oct 9 '13 at 22:33
• @PedroTamaroff well, not exactly using a lot of Fourier series, but deriving using an idea from frequency domain is possible. – peterwhy Oct 9 '13 at 22:36
• @peterwhy (I wouldn't say your answer is using Fourier series, but it is a nice one. +1) – Pedro Tamaroff Oct 9 '13 at 22:39

## 2 Answers

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

Hint: $$2\cos(kx)\,\sin(\frac{x}{2})=\sin\left(kx+\frac{x}{2}\right)- \sin\left(kx-\frac{x}{2}\right)$$