Proofs for two q-trigonometric identities of Gosper

In 2001, Gosper introduced q-analogues for the functions sin⁡z and cos⁡z and stated without proofs many identities involving these q-analogues. Gosper asked whether his formulas are true. In this paper, we shall use the theory of elliptic functions to confirm two of Gosper's identities. Moreover, we shall give two consequences of these identities.