The trigonometric Poschl-Teller (PT) potential describes diatomic molecular vibration. We have obtained the approximate solutions of the radial Schrodinger equation for the rotating trigonometric PT potential using the Nikiforov-Uvarov method. The energy eigenvalues and their corresponding eigenfunctions are calculated for arbitrary l-states in closed form. In the low-screening region, when the screening parameter alpha -> 0, the potential reduces to the Kratzer potential. Numerical results are presented for several diatomic molecules.