  We apply the IC e$^\pm$ pair cascade model in order to investigate the
possible $\gamma$-ray emission features from another compact massive binary of
the microquasar type, Cyg X-1. We conclude that the observational constraints
at lower energies (from MeV to GeV) suggest that the spectrum of electrons
injected in the jet is likely steeper than in other TeV $\gamma$-ray
microquasars. The cascade $\gamma$-ray spectrum produced by electrons with such
spectrum in Cyg X-1 should be below the sensitivities of the MAGIC and VERITAS
class Cherenkov telescopes. However, if the electron spectrum is flatter, then
the highest TeV $\gamma$-ray fluxes are predicted at $\sim$7 hrs before and
after the phase when the compact object is in front of the massive star. We
suggest that Cherenkov telescopes should concentrate on these range of phases
since the TeV flux can vary by a factor of $\sim$20 with the period of the Cyg
X-1 binary system. Moreover, the model predicts clear anticorrelation of the
GeV and TeV $\gamma$-ray emission. This feature can be tested by the future
multiwavelength observations with the AGILE and GLAST telescopes in the GeV
energy range and the MAGIC and VERITAS telescopes in the TeV energy range.
