  We report results of terahertz Faraday and Kerr rotation spectroscopy
measurements on thin films of $\text{Bi}_{1-x}\text{Sb}_{x}$, an alloy system
that exhibits a semimetal-to-topological-insulator transition as the Sb
composition $x$ increases. By using a single-shot time-domain terahertz
spectroscopy setup combined with a table-top pulsed mini-coil magnet, we
conducted measurements in magnetic fields up to 30~T, observing distinctly
different behaviors between semimetallic ($x < 0.07$) and topological insulator
($x > 0.07$) samples. Faraday and Kerr rotation spectra for the semimetallic
films showed a pronounced dip that blue-shifted with the magnetic field,
whereas spectra for the topological insulator films were positive and
featureless, increasing in amplitude with increasing magnetic field and
eventually saturating at high fields ($>$20~T). Ellipticity spectra for the
semimetallic films showed resonances, whereas the topological insulator films
showed no detectable ellipticity. To explain these observations, we developed a
theoretical model based on realistic band parameters and the Kubo formula for
calculating the optical conductivity of Landau-quantized charge carriers. Our
calculations quantitatively reproduced all experimental features, establishing
that the Faraday and Kerr signals in the semimetallic films predominantly arise
from bulk hole cyclotron resonances while the signals in the topological
insulator films represent combined effects of surface carriers originating from
multiple electron and hole pockets. These results demonstrate that the use of
high magnetic fields in terahertz magnetopolarimetry, combined with detailed
electronic structure and conductivity calculations, allows us to unambiguously
identify and quantitatively determine unique contributions from different
species of carriers of topological and nontopological nature in
Bi$_{1-x}$Sb$_x$.
