Homodyne concept of receiver
You can also look at it in the time domain. Your received signal is

$$s(t)=s_I(t)\cos(\omega_c t)+s_Q(t)\sin(\omega_c t)\tag{1}$$

where $s_I(t)$ and $s_Q(t)$ are the in-phase and quadrature components, and $\omega_c$ is the carrier frequency (in rad). We have to assume that at the receiver the carrier frequency and phase are known (coherent demodulation), or can be retrieved somehow.

After demodulation we obtain two signals:

$$\begin{align}s(t)\cos(\omega_ct)&amp;=s_I(t)\cos^2(\omega_ct)+s_Q(t)\sin(\omega_ct)\cos(\omega_ct)\\&amp;=\frac12\big[s_I(t)(1+\cos(2\omega_ct))+s_Q(t)\sin(2\omega_ct)\big]\tag{2}\end{align}$$

and

$$\begin{align}s(t)\sin(\omega_ct)&amp;=s_I(t)\cos(\omega_ct)\sin(\omega_ct)+s_Q(t)\sin^2(\omega_ct)\\&amp;=\frac12\big[s_I(t)\sin(2\omega_ct)+s_Q(t)(1-\cos(2\omega_ct))\big]\tag{3}\end{align}$$

The low pass filters filter out all components at $2\omega_c$, so at the outputs of the filters we're left with

$$r_I(t)=\frac12s_I(t)\tag{4}$$

and

$$r_Q(t)=\frac12s_Q(t)\tag{5}$$