The tangent function and power residues modulo primes

Abstract

Let $p$ be an odd prime, and let $a$ be an integer not divisible by $p$. When $m$ is a positive integer with $p\equiv1\pmod{2m}$ and $2$ is an $m$th power residue modulo $p$, we determine the value of the product $\prod_{k\in R_m(p)}\tan\pi\frac{ak}p$, where $$R_m(p)=\{0<k<p:\ k\in\mathbb Z\ \text{is an}\ m\text{th power reside modulo}\ p\}.$$ In particular, if $p=x^2+64y^2$ with $x,y\in\mathbb Z$, then $$\prod_{k\in R_4(p)}\left(1+\tan\pi\frac {ak}p\right)=(-1)^{y}(-2)^{(p-1)/8}.$$