  The Euclid space telescope will measure the shapes and redshifts of galaxies
to reconstruct the expansion history of the Universe and the growth of cosmic
structures. Estimation of the expected performance of the experiment, in terms
of predicted constraints on cosmological parameters, has so far relied on
different methodologies and numerical implementations, developed for different
observational probes and for their combination. In this paper we present
validated forecasts, that combine both theoretical and observational expertise
for different cosmological probes. This is presented to provide the community
with reliable numerical codes and methods for Euclid cosmological forecasts. We
describe in detail the methodology adopted for Fisher matrix forecasts, applied
to galaxy clustering, weak lensing and their combination. We estimate the
required accuracy for Euclid forecasts and outline a methodology for their
development. We then compare and improve different numerical implementations,
reaching uncertainties on the errors of cosmological parameters that are less
than the required precision in all cases. Furthermore, we provide details on
the validated implementations that can be used by the reader to validate their
own codes if required. We present new cosmological forecasts for Euclid. We
find that results depend on the specific cosmological model and remaining
freedom in each setup, i.e. flat or non-flat spatial cosmologies, or different
cuts at nonlinear scales. The validated numerical implementations can now be
reliably used for any setup. We present results for an optimistic and a
pessimistic choice of such settings. We demonstrate that the impact of
cross-correlations is particularly relevant for models beyond a cosmological
constant and may allow us to increase the dark energy Figure of Merit by at
least a factor of three.
