KöMaL 2017 A697

For all primes $p\ge3$, let $S(p) = \sum_{k=1}^{\frac{p-1}2} \tan \frac{k^2\pi}{p}.$ $(a)$ Show that $p\equiv1\pmod4$ implies $S(p)=0$. $(b)$ Show that if $p\equiv3\pmod4$, then $\dfrac{S(p)}{\sqrt{p}}$ is an odd integer.