Here's how I would do it.
Use the Cosine angle addition formula on $x_1$ and $x_2$:
$$ \cos( \alpha + \beta) = \cos( \alpha ) \cos(\beta)-\sin( \alpha ) \sin(\beta) $$
$$ x_1(t) = a_1 \cos( \omega t + \phi_1 ) $$
$$ x_1 = a_1 \cos( \omega t ) \cos(\phi_1)- a_1 \sin( \omega t ) \sin(\phi_1) $$
The $\omega$s will be the same, so:
$$ x_2 = ...
(1) The phases are indeed relative. The numbers 45, 135, 225 and 315 are relative to the transmitter's oscillator. The receiver, of course, has a different oscillator, so it will need to do some procesing to estimate the phases.
Related to this subject, there is also differential modulation, where 1s and 0s are transmitting by changing the phase relative to ...