  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
$\cos_q z$.
