A Unified Infinite-Series Framework for Finite Trigonometric Sum and Product Identities

We use the Mittag-Leffler expansions of 1/sin2x and 1/sinx, together with the Euler product for sinx, to evaluate finite sums and one finite product of trigonometric functions over the equally spaced points α+kπ/n, k=0,…,n−1. Differentiating one of these expansions l times and summing over the shifted points reduces the finite sum to a rearrangement of a single absolutely convergent series, evaluated at nα; the same argument, applied at α=0, gives closed forms for ∑k=1n−1f(2l)(kπ/n) and its csc-analog in terms of the Riemann zeta function ζ(2l+2) for every nonnegative integer l from one computation. We show plainly that these ζ-linked identities are this paper’s substantive content: the corresponding identities at α≠0 reduce, once their l=0 case is known, to an elementary l-fold differentiation of either an elementary finite identity or an identity already published by Wang, and we do not claim these as new. We also note precisely that our closed forms sum a specific polynomial combination of powers of csc, not an isolated power, and we verify all identities numerically. We discuss the connection to ζ at even integers, one worked application to discrete Fourier transform spectral leakage, and which extensions (tangent/cotangent, hyperbolic) we expect to be routine versus which (elliptic, multidimensional) remain open.