  The data from the Euclid mission will enable the measurement of the
photometric redshifts, angular positions, and weak lensing shapes for over a
billion galaxies. This large dataset will allow for cosmological analyses using
the angular clustering of galaxies and cosmic shear. The cross-correlation (XC)
between these probes can tighten constraints and it is therefore important to
quantify their impact for Euclid. In this study we carefully quantify the
impact of XC not only on the final parameter constraints for different
cosmological models, but also on the nuisance parameters. In particular, we aim
at understanding the amount of additional information that XC can provide for
parameters encoding systematic effects, such as galaxy bias or intrinsic
alignments (IA). We follow the formalism presented in Euclid Collaboration:
Blanchard et al. (2019) and make use of the codes validated therein. We show
that XC improves the dark energy Figure of Merit (FoM) by a factor $\sim 5$,
whilst it also reduces the uncertainties on galaxy bias by $\sim 17\%$ and the
uncertainties on IA by a factor $\sim 4$. We observe that the role of XC on the
final parameter constraints is qualitatively the same irrespective of the
galaxy bias model used. We also show that XC can help in distinguishing between
different IA models, and that if IA terms are neglected then this can lead to
significant biases on the cosmological parameters. We find that the XC terms
are necessary to extract the full information content from the data in future
analyses. They help in better constraining the cosmological model, and lead to
a better understanding of the systematic effects that contaminate these probes.
Furthermore, we find that XC helps in constraining the mean of the
photometric-redshift distributions, but it requires a more precise knowledge of
this mean in order not to degrade the final FoM. [Abridged]
