In this paper we give a classification of left invariant sub-Riemannian structures on cotangent bundle of 2n+1 dimensional Heisenberg group T∗H2n+1. We show that the sub-Riemannian metric is tamed by the corresponding Riemannian metric on T∗H2n+1. We also describe Riemannian and sub-Riemannian geodesics on T∗H2n+1.