Trigonometric identities and quadratic residues

TL;DR

This paper presents new identities involving trigonometric functions, explores products related to quadratic residues modulo primes, and proposes conjectures on related algebraic structures.

Contribution

It introduces novel trigonometric identities for sums involving sine and cosine, and determines specific product values linked to quadratic residues modulo odd primes.

Findings

Derived new identities for sums of trigonometric functions involving complex variables.

Calculated explicit product values for quadratic residues modulo odd primes.

Formulated conjectures on products involving roots of unity 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.