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We know that refractive index, for any medium,
$$n=1/\sqrt{\epsilon\mu}.$$
Also, according to Cauchy's relation
$$n=A+B/\lambda^2,$$
where $A$ and $B$ are constants related to the medium.
According to the first relation, refractive index isn't in any way related to the wavelength of the light, it is only related to the permitivitty and permeability of the medium. According to the second relation, however, it depends on the wavelength too. Which formula is right? Why does the discrepancy arise?
There is no discrepancy. Generally speaking, the permittivity of the medium depends on the wavelength, which means that $n=1/\sqrt{\mu\epsilon}$ also depends on the wavelength. (The magnetic permeability could also depend on the wavelength, but this is much less frequent.)
It's also important to note that Cauchy's relation is only ever an approximation which holds for limited ranges in wavelength, and only ever to a finite precision. As one example, many materials can be modelled very well with the Drude model, which predicts $\epsilon = \epsilon(\omega) = 1-\omega_p^2/\omega^2$, where $\omega_p$ is a parameter that describes the material. For certain ranges of $\omega$ in relationship with $\omega_p$, the Cauchy relation can be a good fit - but even then it is still a phenomenological model that's only a vague description of the rough behaviour of the underlying physics over a limited range of the controlling parameters.
