Note on a simple trigonometric equality

In a recent paper Vinay, Shweta, and Harshitha use Ramanujan’s theta functions to prove several trigonometric equalities. Elementary proofs for some of these equalities were given by the author and Zeilberger, then for all but one by Berndt, Kim, and Zaharescu. The remaining equality was proved in the same paper of Berndt, Kim, and Zaharescu in a non-totally elementary way, in that it uses a result of Gauss in cyclotomy: here we give an elementary proof of the same equality. Also we show how to use automated tools to discover and prove some generalizations.