I am trying to prove that

$\underset{N\to \mathrm{\infty}}{lim}\frac{1}{2N}\sum _{k=1}^{N}[\mathrm{cos}((x-kL)\cdot q)+\mathrm{cos}((x+kL)\cdot q)]=0$

for every $x,L\in {\mathbb{R}}^{+}$ with $x\le L$ and for every $q\ne \frac{2n\pi}{L}$, $n\in \mathbb{Z}$.

$\underset{N\to \mathrm{\infty}}{lim}\frac{1}{2N}\sum _{k=1}^{N}[\mathrm{cos}((x-kL)\cdot q)+\mathrm{cos}((x+kL)\cdot q)]=0$

for every $x,L\in {\mathbb{R}}^{+}$ with $x\le L$ and for every $q\ne \frac{2n\pi}{L}$, $n\in \mathbb{Z}$.