The tangent function and power residues modulo primes

Let $p$ be an odd prime, and let $a$ be an integer not divisible by $p$. When $m$ is a positive integer with $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}2m)$ 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 $$R_m(p)=\{0<k<p:k\in \mathbb{Z}\text{ is an }m\text{th power reside modulo }p\}\mathrm{.}$$ In particular, if $p=x^2+64y^2$ with $x,y\in \mathbb{Z}$, then $$\prod _{k\in R_4(p)}(1+\tan \pi \frac{ak}{p})=(-1{)}^y(-2{)}^{(p-1)\mathrm{/}8}\mathrm{.}$$