In an interesting article entitled “Experiments and discoveries in $q$-trigonometry”, R. W. Gosper conjectured few beautiful $\Pi_q$ Lambert series identities. Many people have attempted confirming some of those identities the Gosper’s list, mainly by using $q$-trigonometric this paper we either prove or disprove all list S. Ramanujan’s theta function N. Bailey’s summation formula. process, obt...