  We investigate how precisely the CP nature of the 125 GeV Higgs boson,
parametrized by a scalar-pseudoscalar Higgs mixing angle, can be determined in
Higgs-to-tau-pair decay with subsequent tau-lepton decays to charged prongs at
the Large Hadron Collider (LHC). We combine two methods in order to define an
observable which is sensitive to this scalar-pseudoscalar mixing angle: We use
the rho-decay plane method for tau-to-rho decays and the impact parameter
method for all other major tau decays. For estimating the precision with which
the mixing angle can be measured at the LHC (13 TeV) we take into account the
background from Drell-Yan production and perform a Monte Carlo simulation of
measurement uncertainties on the signal and background distributions. We obtain
that the mixing angle can be determined with an uncertainty of 15 degree (9
degree) at the LHC with an integrated luminosity of 150 inverse fb (500 inverse
fb), and with ~4 degree with 3 inverse ab. Future measurements of the
scalar-pseudoscalar mixing angle yield direct information on whether or not
there is an extended Higgs-boson sector with Higgs-sector CP violation. We
analyze this in the context of a number of two-Higgs-doublet extensions of the
Standard Model, namely the so-called aligned model and conventional
two-Higgs-doublet extensions with tree-level neutral flavor conservation.
