I'm trying to prove that $X$ (given in the title of the question) is a Killing vector field for a metric which is spherically symmetric and static.
This is what I've done so far:
The vector field $X$ in the basis $(\partial_t, \partial_r, \partial_\theta, \partial_\phi)$ has components $(0,0,\sin \phi, \cot \theta \cos \phi)$ and the metric only depends on $r$, $g_{\mu \nu} = g_{\mu \nu} (r)$.
$X$ is Killing if the Lie derivative of the metric with respect to the field is 0, $\mathcal{L}_X g = 0$. I'm going to use this expression in components to try to prove the result.
$$(\mathcal L_X g)_{\mu \nu} = X^\rho \partial_\rho g_{\mu \nu} + g_{\mu \nu} \partial_\nu X^\rho + g_{\mu \nu} \partial_\mu X^\rho$$
In this case the only possible non-zero components are those for $\mu, \nu = \theta, \phi$ and I get the following equation for arbitrary $\mu \nu$:
$$g_{\mu \theta} \partial_\nu \sin \phi + g_{\theta \nu} \partial_\mu \sin \phi + g_{\mu \phi} \partial_\nu (\cot \theta \cos \phi) + g_{\phi \nu} \partial_\mu (\cot \theta \cos \phi)$$
But I don't know how to prove that this is 0 for all values of $\mu$ and $\nu$ (for example I've been having problems with $\mu=\phi$ and $\nu=\theta$). How can I show that this is 0?