I came across this summation while solving harmonic chain.
\begin{align} \sum_{l=1}^N \sin^2{(\frac{m\pi}{N+1}\cdot l)} \end{align} for an integer m that can range from 1 to N.
So the question is if the value of the summation changes with m? I checked for until N=5. And it remains the same for all integer m between 0 and N+1. So I'm wondering if the pattern continues for all integer values of N and if so is there a mathematical proof someone can provide for the same.
Edit: I also found that the summation evaluates to $\frac{N+1}{2}$ for all N upto 5. which also makes sense as when N goes to infinity this becomes an integral over the index $l$ that evaluates to the same value $\frac{N+1}{2}$
Edit 2: I tried again to check for higher values of N and wrote a program for that and I checked for until N=500. And this still works perfectly. I don't know why I couldn't find a proof until now. As I know it works, I will try to see an inductive proof if possible.