Proof of Gosper's sine of sum of n angles identity

Proof of Gosper’s sine of sum of n angles identity

$sin(\sum_{k=1}^n x_k)=\frac{2^{n-1}}{n}\sum_{k=1}^n (-1)^{k-1} \prod_{j=1}^n sin(x_j-\frac{(k-j)\pi}{n})$ Gosper published very few papers about combinatory and hypergeometric sums, but none resource reveals the derivation of this sine identity. Any derivation hint/link help is appreciated!

flag	asked Jun 24 at 18:18 J Chu 1

Write $e(x)=e^{ix}$; expand $\sin(x)$ as $(e(x)−e(−x))/(2i)$; expand the product by the distributive law as a sum of terms of the form $\sum_\epsilon(\sum_k a_{\epsilon k} e(\sum \epsilon_jx_j))$ (where $\epsilon$ runs over elements of $\lbrace \pm 1\rbrace^n$) and make sure that all of the terms sum to 0 except the ones with all $+$'s and all $-$s. – Anthony Quas Jun 24 at 18:57