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 $$\sum _{r=0}^{n-1}\frac{1}{1+\sin 2\pi \frac{x+r}{n}+\cos 2\pi \frac{x+r}{n}}=\frac{(-1{)}^{(n-1)/2}n}{1+(-1{)}^{(n-1)/2}\sin 2\pi x+\cos 2\pi x}$$ for any complex number with $x+1/2,x+(-1{)}^{(n-1)/2}/4\notin \mathbb{Z}$, and $$\sum _{j,k=0}^{n-1}\frac{1}{\sin 2\pi \frac{x+j}{n}+\sin 2\pi \frac{y+k}{n}}=\frac{(-1{)}^{(n-1)/2}{n}^2}{\sin 2\pi x+\sin 2\pi y}$$ for all complex numbers $x$ and $y$ with $x+y,x-y-1/2\notin \mathbb{Z}$. 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 $G_p(x)={\prod }_{k=1}^{(p-1)/2}(x-e^{2\pi i{k}^2/p})$ with $p$ an odd prime and $x$ a root of unity.

arXiv comment27 pages. Accepted by Publ. Math. Debrecen