Given the following equation
\begin{equation} g_{\alpha\delta} \Gamma^{\delta}_{\beta\gamma} = \frac{1}{2} \left(\partial_\gamma g_{\alpha\beta} + \partial_\beta g_{\alpha\gamma} - \partial_\alpha g_{\beta\gamma} \right) \end{equation}
Suppose I have the following metric
\begin{equation} g_{\alpha\beta} = \begin{bmatrix} a^2 & 0 \\ 0 & a^2\sin ^2 \theta \\ \end{bmatrix} \end{equation}
$\alpha,\beta \in \lbrace \theta, \phi \rbrace $
So suppose I want to find all the Christoffel symbols for this metric.
Work I Did:
\begin{align*} g_{\theta\theta} \Gamma^\theta_{\theta\theta} & = 0\\ g_{\theta\theta} \Gamma^\theta_{\theta\phi} & = 0\\ g_{\theta\theta} \Gamma^\theta_{\phi\phi} & = -\sin\theta\cos\theta\\ g_{\phi\phi} \Gamma^\phi_{\theta\theta} & = 0\\ g_{\phi\phi} \Gamma^\phi_{\theta\phi} & = \cot \theta \\ g_{\phi\phi} \Gamma^\phi_{\phi\phi} & = 0\\ \end{align*}
But here's the problem, suppose I just want to find $\Gamma^\phi_{\theta\phi} $ by itself, can I just divide $\cot \theta$ by $g_{\phi\phi}$?