  We explore connections between the electronic density of states (DOS) in a
conducting system and the frequency dependence of the scattering rate
$1/\tau(\omega)$ inferred from infrared spectroscopy. We show that changes in
the DOS upon the development of energy gaps can be reliably tracked through the
examination of the $1/\tau(\omega)$ spectra using the sum rules discussed in
the text. Applying this analysis to the charge dynamics in high-$T_c$ cuprates
we found radically different trends in the evolution of the DOS in the
pseudogap state and in the superconducting state.
