On q-analogues for trigonometric identities

Abstract

AbstractIn this paper we will give q-analogues for the Pythagorean trigonometric identity {\sin^{2}z+\cos^{2}z=1} in terms of Gosper’s q-trigonometry. We shall also give new q-analogues for the duplicate trigonometric identity {\sin(x-y)\sin(x+y)=\sin^{2}x-\sin^{2}y}. Moreover, we shall give a short proof for an identity of Gosper, which was also established by Mező. The main argument of our proofs is the residue theorem applied to elliptic functions.