  Future data provided by the Euclid mission will allow us to better understand
the cosmic history of the Universe. A metric of its performance is the
figure-of-merit (FoM) of dark energy, usually estimated with Fisher forecasts.
The expected FoM has previously been estimated taking into account the two main
probes of Euclid, namely the three-dimensional clustering of the spectroscopic
galaxy sample, and the so-called 3x2pt signal from the photometric sample
(i.e., the weak lensing signal, the galaxy clustering, and their
cross-correlation). So far, these two probes have been treated as independent.
In this paper, we introduce a new observable given by the ratio of the
(angular) two-point correlation function of galaxies from the two surveys. For
identical (normalised) selection functions, this observable is unaffected by
sampling noise, and its variance is solely controlled by Poisson noise. We
present forecasts for Euclid where this multi-tracer method is applied and is
particularly relevant because the two surveys will cover the same area of the
sky. This method allows for the exploitation of the combination of the
spectroscopic and photometric samples. When the correlation between this new
observable and the other probes is not taken into account, a significant gain
is obtained in the FoM, as well as in the constraints on other cosmological
parameters. The benefit is more pronounced for a commonly investigated modified
gravity model, namely the $\gamma$ parametrisation of the growth factor.
However, the correlation between the different probes is found to be
significant and hence the actual gain is uncertain. We present various
strategies for circumventing this issue and still extract useful information
from the new observable.
