  In this article, we study Prasad's conjecture for regular supercuspidal
representations based on the machinery developed by Hakim and Murnaghan to
study distinguished representations, and the fundamental work of Kaletha on
parameterization of regular supercuspidal representations. For regular
supercuspidal representations, we give some new interpretations of the
numerical quantities appearing in Prasad's formula, and reduce the proof to the
case of tori. The proof of Prasad's conjecture then reduces to a comparison of
various quadratic characters appearing naturally in the above process. We also
have some new observations on these characters and study the relation between
them in detail. For some particular examples, we show the coincidence of these
characters, which gives a new purely local proof of Prasad's conjecture for
regular supercuspidal representations of these groups. We also prove Prasad's
conjecture for regular supercuspidal representations of G(E), when E/F is
unramified and G is a general quasi-split reductive group.
