  We evaluate the flux of tau neutrinos produced from the decay of pair
produced taus from incident muons using a cascade equation analysis. To solve
the cascade equations, our numerical result for the tau production $Z$ moment
is given. Our results for the flux of tau neutrinos produced from incident
muons are compared to the flux of tau neutrinos produced via oscillations and
the direct prompt atmospheric tau neutrino flux. Results are given for both
downward and upward going neutrinos fluxes and higher zenith angles are
discussed. We conclude that the direct prompt atmospheric tau neutrino flux
dominates these other atmospheric sources of tau neutrinos for neutrino
energies larger than a few TeV for upward fluxes, and over a wider range of
energy for downward fluxes.
