transmitting the real/imaginary parts of pass-band signal
Will we be able to recover the complex modulated symbol based on $s_2(t)$ too?

Yes, no information is lost between the two.

How can we prove that mathematically?

One way to think about it is to write out each individually and see what you have. Let $a(t)=a_I(t)+ja_Q(t)$, expanding $s_1(t)$ and $s_2(t)$ we have:
\begin{align}
s_1(t) &amp;= a_I(t)\text{cos}(2\pi f_ct)-a_Q(t)\text{sin}(2\pi f_ct) \\
s_2(t) &amp;= a_I(t)\text{sin}(2\pi f_ct)+a_Q(t)\text{cos}(2\pi f_ct)
\end{align}
The in-phase and quadrature parts are present in each so we just need to bring them back to baseband to get $a(t)$ back. Further, whatever workings you have to do this for $s_1(t)$ you can reuse for the processing of $s_2(t)$ by first applying a simple $+\frac{\pi}{2}$ to it ($\text{cos}(\theta + \frac{\pi}{2})=-\text{sin}(\theta)$).