Prove that $\sum ^{n} _{k=1} \sin(kx) \le \frac{1}{|\sin(\frac{x}{2})|}$ 
Prove that $\sum ^{n} _{k=1} \sin(kx) \le \frac{1}{|\sin(\frac{x}{2})|}$

I am doing some task and I wanted to see solution which is in my book. However there is inequality $\sum ^{n} _{k=1} \sin(kx) \le \frac{1}{|\sin(\frac{x}{2})|}$ and I don't understand why it is true.Author of this inequality wanted to show that a sequence of partial sums series $ \sin(nx)$ is uniformly limited.Can you help me?

$\sum\limits_{k=1}^n\sin{kx}\sin{\frac{x}{2}}=\frac{1}{2}\sum\limits_{k=1}^n(\cos(kx-\frac{x}{2})-\cos(kx+\frac{x}{2}))=\frac{1}{2}(\cos(\frac{x}{2})-\cos(\frac{3x}{2})+\cos(\frac{3x}{2})-\cos(\frac{5x}{2})+...)=\frac{1}{2}(\cos{\frac{x}{2}}-\cos\frac{(2n+1)x}{2})$
So $|\sum\limits_{k=1}^n\sin{kx}|\leq \frac{\frac{1+1}{2}}{|\sin{\frac{x}{2}}|}= \frac{1}{|\sin{\frac{x}{2}}|}$
