I have a trigonometric sum as below $$\frac{1}{N^2}\sum_{r=0}^{N-1}\frac{\sin^2(\pi e)}{\sin^2(\frac{\pi(r-n+e))}{N})}\frac{\cos^2(\frac{\pi(Ne-e-r+n)}{N})}{\cos^2(\frac{\pi(Ne-e)}{N})}$$ and I want to show analytically that for small $e$ ($e<0.2$) and large $N$ the above sum approximately equals $1$. I have showed this numerically in MATLAB but I cant show it analytically. By the way $n$ is an integer between $0$ and $N-1$.
Can any one help?
Thanks in advance.