Trigonometric identities and quadratic residues

Abstract

In this paper we obtain some novel identities involving trigonometric functions. Let $n$ be any positive odd integer. We show that $$\sum_{r=0}^{n-1}\frac1{1+\sin2\pi\frac{x+r}n+\cos2\pi\frac{x+r}n} =\frac{(-1)^{(n-1)/2}n}{1+(-1)^{(n-1)/2}\sin 2\pi x+\cos 2\pi x}$$ for any complex number with $x+1/2,x+(-1)^{(n-1)/2}/4\not\in\mathbb Z$, and $$\sum_{j,k=0}^{n-1}\frac1{\sin 2\pi\frac{x+j}n+\sin2\pi \frac{y+k}n}=\frac{(-1)^{(n-1)/2}n^2}{\sin 2\pi x+\sin2\pi y}$$ for all complex numbers $x$ and $y$ with $x+y,x-y-1/2\not\in\mathbb Z$. We also determine the values of $\prod_{k=1}^{(p-1)/2}(1+\tan\pi\frac{k^2}p)$ and $\prod_{k=1}^{(p-1)/2}(1+\cot\pi\frac{k^2}p)$ for any odd prime $p$. In addition, we pose several conjectures on the values of $G_p(x)=\prod_{k=1}^{(p-1)/2}(x-e^{2\pi ik^2/p})$ with $p$ an odd prime and $x$ a root of unity.