Is this system of ODE's exactly solvable?

{\sin^2(\theta(t))\dot{\phi}(t)=C}\brace{\sin(\the ta(t)\cos(\theta(t))\dot{\phi}^2(t)=\ddot{\theta}( t)}

Notice that above the \phi is a dot which of course is meant to represent \frac{\partial}{\partial t}

If anyone's curious it arose from trying to find the extremals of the functional J\left(\theta,\phi\right)=\int\left(\left\|\frac{\ partial \vec{x}}{\partial t}\right\|^2-U(t)\right)dt where in this case U(t)=0 and we're assuming that \vec{x}\in\mathbb{S}^2