# Proof of an inequality problem

Wise men or women over the world!! I badly ask you to help me.

Let $N$ and $B$ be two positive integers such that $1\le B\le \frac{N}{2}$ and $N=ug$ (for convenience, assume that $N$ is even.)

For instance, $N=20$, $u=4$, $g=5$, $1\le B\le 5$ (of course, $u$ and $g$ can be variant).

I want to prove that

$$\sum_{m=0}^{u-1} \csc \left(\frac{\pi}{N}(a+mg) \right) \left| \sin \left(\frac{\pi}{N}B(a+mg) \right) \right| \leq B+ \sum_{m=1}^{u-1} \csc(\frac{\pi}{N}mg),$$

where $1 \leq a \leq g-1$, for any $u, g$ such that $N=ug$, and $1\le B \le \frac{N}{2}$.

As you know, $\csc(x) =\frac{1}{\sin(x)}$.

Seemingly, it may be true because $|\sin((\frac{\pi}{N})B(a+mg))|$ in the left-hand side can not be greater than 1 but I hope to prove it mathematically.

May Anybody help me (prove it or tell me the statement is not true otherwise).

I stay up this night waiting for your kindness!!

-