Sum of sines inequality

Question

I need to prove the following inequality: $$\bigg\lvert \sum_{n=1}^{N}\sin(nx)\bigg\rvert \leq \frac{1}{\sin(x/2)}, \, x\neq 2k\pi,k\in \mathbb{Z}$$ No idea where to start. Any tips?

Answer 1

Hint:

You can start with the factorisation formula (or prove it – it's a standard high-school exercise): $$ \sum_{n=1}^{N}\sin(nx)=\frac{\sin\frac{(N+1)x}2}{\sin\frac x2}\,\sin \frac{Nx}2.$$

Answer 2

Hint.

$$ \sum_{n=1}^N \sin(n x)= \mbox{Im}\left[\sum_{n=1}^N e^{i n x }\right] = \mbox{Im}\left[\frac{e^{i(N+1)x}-1}{e^{i x}-1}\right] $$