A communication system
Let me slightly change notation and denote the two modulation frequencies by $\omega_1$ and $\omega_2$.
You need $|\omega_1-\omega_2|&gt;2\omega_{max}$ to be satisfied, otherwise the shifted spectral of $x(t)$ and $y(t)$ will overlap. If this condition is satisfied, you can retrieve $x(t)$ by demodulation (multiplication with $\cos(\omega_1t)$) and (scaled) low pass filtering, as you suggested:
$$\begin{align}2x(t)\cos^2(\omega_1t)=x(t)\big[1+\cos(2\omega_1t)\big]\Big|_{low pass}=x(t)\end{align}\tag{1}$$