Let $T_{p}(n,m) = \displaystyle{\frac{\sin( \frac{ p \pi}{n}) }{ \sin(\frac{\pi}{n})}-\frac{m+\sqrt{m^2+4}}{2}}$ for $p,n,m$ integers.
When is $T_{p}(n,m) = 0$ ? , for instances for $p=2,n=5,m=1$ and for $p=3 , n=8 ,m=2$ we get that the function $T$ is $0$.
But is there any other values for which $T =0$ ?!!
Edit : $2\leq p < n-1$ and $m$ is positive.
Update : question posted on MO
Regarding your question (are there any other values?): Yes, $p=3, n=5, m=1$ and $p=5, n=8, m=2$, or am I missing anything obvious?