# A new trigonometric identity with applications

## TL;DR

This paper introduces a novel trigonometric identity involving sums of cotangent and sine functions for odd integers, with applications to Chebyshev polynomials and Bernoulli polynomials, revealing new mathematical relationships.

## Contribution

It presents a new trigonometric identity for odd integers and explores its implications for Chebyshev and Bernoulli polynomials, expanding mathematical understanding.

## Key findings

- Proves a new identity involving trigonometric sums for odd integers.
- Derives equivalent identities involving Chebyshev polynomials.
- Establishes zero-sum identities for Bernoulli polynomials with specific parameters.

## Abstract

In this paper we obtain a new curious identity involving trigonometric functions. Namely, for any positive odd integer $n$ we prove that $$\sum_{k=1}^n(-1)^k(\cot kx)\sin k(n-k)x=\frac{1-n}2,$$ which is equivalent to the identity $$\sum_{k=1}^n(-1)^kU_{n-k}(\cos kx)=-\frac{n+1}2,$$ where $U_m(z)$ stands for the $m$th Chebyshev polynomial of the second kind. As a consequence, for any positive odd integer $n$ and positive integer $m$ we obtain $$\sum_{k=1}^n(-1)^kk^{2m}B_{2m+1}\left(\frac{n-k}2\right)=0,$$ where $B_j(x)$ denotes the Bernoulli polynomial of degree $j$.