I have the following sum:
$$\sum_{\sigma=1,\text{odd}}^{\frac{p}{2}-2}\frac{\sin \frac{r \sigma \pi}{p}}{\sin \frac{q \sigma \pi}{p}}$$
where $p\equiv2\pmod4$, $p$ and $q$ are coprime numbers such that $p-q\ge1$, and $r\in \{1,\dots,p-1\}$ is odd. I know the result is of the form $$\pm \frac{\frac{p}{2}\pm 1}{2},$$ but the signs depend on the relation of $r, p, q$, and it is not easy to guess.
Is there some formula for this type of sums?
I tried to reduce it using exponentials as the following:
$$\sum_{\sigma=1,odd}^{\frac{p}{2}-2}e^{i\pi\frac{(q-r)\sigma}{p}}\frac{e^{2i\pi\frac{r\sigma}{p}}-1}{e^{2i\pi\frac{q\sigma}{p}}-1}$$
and define $\omega=e^{2 i\pi\frac{q}{p}}$ which is a primitive root of unity but I don't know how this can help.
Please help! Thanks!!!
