I am working on a problem, and came across the following expression after some manipulations:
$\cfrac{sin(\frac{\pi k}{m})}{sin(\frac{\pi}{m})} - \cfrac{sin(\frac{\pi}{m})}{sin(\frac{\pi k}{m})} $.
To find an answer to my question, I would need to know for what positive integers $k$ and $m$ (with $k<m$) this expression is a positive integer (or that there aren't any besides in 2 cases), two known solutions are $k=2$, $m=5$, and $k=3$, $m=8$.
I have spent much time thinking about the question, and I feel as if I am lacking some key idea necessary to try to answer this question. Are there any methods for determining when this expression is a positive integer?