Proving some identities of Gosper on q-trigonometric functions

Gosper introduced the functions $\sin_q z$ and $\cos_q z$ as $q$-analogues for the trigonometric functions $\sin z$ and $\cos z$ respectively. He stated but did not prove a variety of identities involving these two $q$-trigonometric functions. In this paper, we shall use the theory of elliptic functions to prove three formulas from the list of Gosper on the functions $\sin_q z$ and $\cos_q z$.