## The zeroes of a tricky function

I'm trying to prove that the function

$\displaystyle h(x)=\sum_{n=2}^{trunc(\sqrt(x))} \frac{\cos(\frac{\pi x}{n})}{n\sin(\frac{\pi x}{n})}+\frac{\cos(\frac{(x+2)\pi}{n})}{n\sin(\fra c{(x+2)\pi}{n})}$

does not have a zero on every unit $(k,k+1), k\in\mathbb{N}$ for large enough k. Any ideas?

Moderator edit: To the OP, please use math tags for the latex. I have edited your post accordingly.