A new trigonometric identity with applications

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 $$\sum _{k=1}^n(-1{)}^k{U}_{n-k}(\cos kx)=-\frac{n+1}{2},$$ 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 $$\sum _{k=1}^n(-1{)}^k{k}^{2m}{B}_{2m+1}\left(\frac{n-k}{2}\right)=0,$$ where $B_j(x)$ denotes the Bernoulli polynomial of degree $j$.

6 pages