In quantum mechanics a localized attractive potential typically supports a (possibly infinite) set of bound states, characterized by a discrete spectrum of allowed energies, together with a continuum of scattering states, characterized (in one dimension) by energy-dependent phase shift. The Morse potential named after physicist Philip M.Morse, is a convenient model for the potential energy of a diatomic molecule. It is a better approximation for the vibrational structure of the molecule than the quantum harmonic oscillator because it explicitly includes the effects of bond breaking such as the existence of unbound states. It also accounts for the anharmonicity of real bonds and the non-zero transition probability for overtone and combination bands. The Morse potential can also be used to model other interactions such as the interaction between an atom and a surface. We here make the case that the trigonometric Rosen-Morse potential is exactly soluble in terms of a family of real orthogonal polynomials and present the solutions. and analytical solutions of the Klein-Gordon equation for the Rosen-Morse potential with equal scalar and vector potentials are studied in this paper.