Human and automated approaches for finite trigonometric sums

Abstract

We show that identities involving trigonometric sums recently proved by Harshitha, Vasuki and Yathirajsharma, using Ramanujan's theory of theta functions, were either already in the literature or can be proved easily by adapting results that can be found in the literature. Also we prove two conjectures given in that paper. After mentioning many other works dealing with identities for various trigonometric sums, we end this paper by describing an automated approach for proving such trigonometric identities. L := sin 4π 13 sin 2π 13 sin 6π 13 sin 3π 13 − sin 2π 13 sin π 13 sin 3π 13 sin 5π 13 − sin 5π 13 sin 4π 13 sin π 13 sin 6π 13 which can also be written, using the relations sin x = sin(π−x), sin(2x) = 2 sin x cos x, 2 cos x cos y = 10 cos(x + y) + cos(x − y) and cos(π − x