On Gosper's Pi(q) and Lambert series identities

In an interesting article entitled `Experiments and discoveries in q-trigonometry'', R. W. Gosper conjectured few beautiful Pi(q) and Lambert series identities. Many people have attempted confirming some of those identities in the Gosper's list, mainly by using Gosper's q-trigonometric identities. In this paper we either prove or disprove all the Pi(q) and Lambert series identities in the Gosper's list by mainly using S. Ramanujan's theta function identities and W. N. Bailey's summation formula. In the process, we obtain three new Gosper kind of identities.