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

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

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