On the Gosper.s q-constant .q

Gosper introduced the functions sinqz and cosqz as q-analogues for the trigonometric functions sin z and cos z respectively. He stated a variety of identities involving these two q-trigonometric functions along with certain constants denoted by .qn (n . N). Gosper noticed that all his formulas on these constants have more than two of the .qn. So, it is natural to raise the question of establishing identities involving only two of the .qn. In this paper, our main goal is to give examples of such formulas in only two .qn.