The coupled propagation of two electromagnetic waves in plasma is studied to establish the conditions for induced transparency. Induced transparency refers to the situation where both waves propagate unattenuated, although the frequency of one (or both) of them is below the plasma frequency so that it could not propagate in the absence of the other. The effect is due to the interaction of the waves through their beat, which modulates both the electron mass and, by exciting longitudinal plasma oscillations, their number density, and thus the plasma frequency. Starting from a relativistic fluid description, a dispersion relation for plane waves of weakly relativistic intensities is derived, which takes into account the polarization of the waves and the nonlinearities with respect to both their amplitudes. This serves as a basis for the exploration of the conditions for induced transparency and the modes of propagation.