I would greatly appreciate help solving this system in Mathematica.
\begin{aligned}\frac{d\theta_{1}}{dt}&=\frac{G(r_s^{2}-1)}{4(r_s^{2}+1)}\sin2\theta_{1}\sin2\phi_{1},\\\\\frac{d\phi_{1}}{dt}&=\frac{G}{{r_s^{2}+1}}\left(r_{s^{2}}\cos^{2}\phi_{1}+\sin^{2}\phi_{1}\right),\end{aligned}
The answer:
\begin{aligned}\tan\theta_{1}&=\frac{Cr_{s}}{({r_{s}}^{2}\cos^{2}\phi_{1}+\sin^{2}\phi_{1})^{\frac{1}{2}}},\\\tan\phi_{1}&=r_{s}\tan\left(\frac{2\pi t}{T}+\kappa\right),\end{aligned} Where T:
\begin{aligned}T=\frac{2\pi}{G}\left(r_{s}+r_{s}^{-1}\right)\end{aligned}
Here $G, r_s$ are constants and $C, k$ are integration constants.
Clear[theta, phi, t, G, rS]
eq1 = D[theta[t],t] == (G (rS^2 - 1))/(4 (rS^2 + 1)) Sin[2 theta[t]] Sin[2 phi[t]];
eq2 = D[phi[t], t] == G/(rS^2 + 1) (rS^2 Cos[phi[t]]^2 +Sin[phi[t]]^2);
solutions = DSolve[{eq1, eq2}, {theta[t], phi[t]}, t];
thetaSol = theta[t] /. solutions[[1]];
phiSol = phi[t] /. solutions[[1]];
simplifiedThetaSol = Simplify[thetaSol]
simplifiedPhiSol = Simplify[phiSol]
I attached this simple code I wrote. The equation I am looking is initial condition dependent (Later I want to do some statistical analysis on this dependency, therefore I need analytical expression.). I wanted to see if MMA can handle this type of equations to reduce work. It gave good result for $\phi$ as it is straightforward to integrate. But once you plug it back to $\theta$ and perform integration it fails. I also tried other symbolic solvers without results.