  Dark matter trapped in the Sun produces a flux of all flavors of neutrinos,
which then reach the Earth after propagating out of the Sun and oscillating
from the production point to the detector. The typical signal which is looked
at refers to the muon neutrino component and consists of a flux of up-going
muons in a neutrino detector. We propose instead a novel signature: the
possibility of looking at the tau neutrino component of the dark matter signal,
which is almost background-free in the downward-going direction, since the tau
neutrino amount in atmospheric neutrinos is negligible and in the down-going
baseline atmospheric muon-neutrinos have no time to sizably oscillate. We
analyze the prospects of studying the downward-going tau neutrinos from dark
matter annihilation (or decay) in the Sun in Cherenkov detectors, by looking at
hadronic showers produced in the charged-current tau neutrino interactions and
subsequent tau decay. We discuss the various sources of background (namely the
small tau neutrino component in atmospheric neutrinos, both from direct
production and from oscillations; tau neutrinos from solar corona interactions;
the galactic tau neutrino component) as well as sources of background due to
misidentification of electron and muon events. We find that the downward-going
tau neutrinos signal has potentially very good prospects for Mton scale
Cherenkov detectors, the main limitation being the level of misidentification
of non-tau events, which need to be kept at level of percent. Several tens of
events per year (depending on the dark matter mass and annihilation/decay
channel) are potentially collectible with a Mton scale detector, and a 5 sigma
significance discovery is potentially reachable for dark matter masses in the
range from 20 to 300 GeV with a few years of exposure on a Mton detector.
