  We develop a theory of the magneto-optical and magneto-electric properties of
a topological insulator thin film in the presence of a quantizing external
magnetic field. We find that low-frequency magneto-optical properties depend
only on the sum of top and bottom surface Dirac-cone filling factors
$\nu_{\mathrm{T}}$ and $\nu_{\mathrm{B}}$, whereas the low-frequency
magneto-electric response depends only on the difference. The Faraday rotation
is quantized in integer multiples of the fine structure constant and the Kerr
effect exhibits a $\pi/2$ rotation. Strongly enhanced cyclotron-resonance
features appear at higher frequencies that are sensitive to the filling factors
of both surfaces. When the product of the bulk conductivity and the film
thickness in $e^2/h$ units is small compared to $\alpha$, magneto-optical
properties are only weakly dependent on accidental doping in the interior of
the film.
