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One outstanding problem in extrasolar planet studies is why no co-orbital exoplanets have been found, despite numerous searches among the many known planetary systems, many of them in other mean-motion resonances. Here we examine the hypothesis that dissipation of energy by tides in Trojan planets is preventing their survival. The Appendix of this paper generalizes the conventional theory of tides to include tidal forces independent of dissipation, as well as the effects of one body on tides raised by another. The main text applies this theory to a model system consisting of a primary of stellar mass, a secondary of sub-stellar mass in a circular orbit about the primary, and a much lighter Trojan planet librating with small amplitude about an equilateral point of the system. Next, we linearize the equations of motion about the Trojan points, including the tidal forces, and solve for the motion of the Trojan. The results indicate that tides damp out the Trojan's motion perpendicular to the orbital plane of the primary and secondary, as well as its epicycles due to its eccentricity; but they pump up the amplitude of its tadpole librations exponentially. We then verify our analytic solutions by integrating the non-linearized equations of motion numerically for several sample cases. In each case, we find that the librations grow until the Trojan escapes its libration, which leads to a close encounter with either the primary or the secondary.