  Using the path-integral approach developed by Br\'ezin et al. [
Nucl.Phys.B235, 24 (1984) ], we obtain an analytical expression for the density
of states of a spin-1/2 disordered two-dimensional electron gas in a strong,
perpendicular magnetic field. The density of states of this system illustrates
the interplay between the Zeeman splitting of Landau levels and the
disorder-induced broadening. We find that the broadening and the band splitting
of the Landau bands are enhanced due to the level repulsion from the mixing of
two spin orientations caused by the random scatterings. The comparison between
the spin-1/2 model and the double-layer system is also discussed.
