  Amongst standard model parameters that are constrained by cosmic microwave
background (CMB) observations, the optical depth $\tau$ stands out as a
nuisance parameter. While $\tau$ provides some crude limits on reionization, it
also degrades constraints on other cosmological parameters. Here we explore how
21 cm cosmology---as a direct probe of reionization---can be used to
independently predict $\tau$ in an effort to improve CMB parameter constraints.
We develop two complementary schemes for doing so. The first uses 21 cm power
spectrum observations in conjunction with semi-analytic simulations to predict
$\tau$. The other uses global 21 cm measurements to directly constrain low
redshift (post-reheating) contributions to $\tau$ in a relatively
model-independent way. Forecasting the performance of the upcoming Hydrogen
Epoch of Reionization Array, we find that significant reductions in the errors
on $\tau$ can be achieved. These results are particularly effective at breaking
the CMB degeneracy between $\tau$ and the amplitude of the primordial
fluctuation spectrum $A_s$, with errors on $\ln (10^{10} A_s)$ reduced by up to
a factor of four. Stage 4 CMB constraints on the neutrino mass sum are also
improved, with errors potentially reduced to $12\,\textrm{meV}$ regardless of
whether CMB experiments can precisely measure the reionization bump in
polarization power spectra. Observations of the 21 cm line are therefore
capable of improving not only our understanding of reionization astrophysics,
but also of cosmology in general.
