Given two points on a sphere $(x_1,y_1,z_1)=(\sin \phi_1 \cos \theta_1,\sin \phi_1 \sin \theta_1,\cos \phi_1 )$ and $(x_2,y_2,z_2)=(\sin \phi_2 \cos \theta_2,\sin \phi_2 \sin \theta_2,\cos \phi_2 )$. Take the dot product of these to obtain the cosine $\alpha$ of the angle between them relative to the center of the sphere. We have \begin{eqnarray*} \cos \alpha = \cos \phi_1 \cos \phi_2 +\sin \phi_1 \sin \phi_2 ( \cos \theta_1 \cos \theta_2 +\sin \theta_1 \sin \theta_2) \\ \cos \alpha = \cos \phi_1 \cos \phi_2 +\sin \phi_1 \sin \phi_2 \cos (\theta_1 - \theta_2 ). \end{eqnarray*}

Given two points on a sphere $(x_1,y_1,z_1)=(\sin \phi_1 \cos \theta_1,\sin \phi_1 \sin \theta_1,\cos \phi_1 )$ and $(x_2,y_2,z_2)=(\sin \phi_2 \cos \theta_2,\sin \phi_2 \sin \theta_2,\cos \phi_2 )$. Take the dot product of these to obtain the cosine of the angle $\alpha$ between them relative to the center of the sphere. We have \begin{eqnarray*} \cos \alpha = \cos \phi_1 \cos \phi_2 +\sin \phi_1 \sin \phi_2 ( \cos \theta_1 \cos \theta_2 +\sin \theta_1 \sin \theta_2) \\ \cos \alpha = \cos \phi_1 \cos \phi_2 +\sin \phi_1 \sin \phi_2 \cos (\theta_1 - \theta_2 ). \end{eqnarray*} \begin{eqnarray*} \alpha = \color{blue}{\cos^{-1} \left( \cos \phi_1 \cos \phi_2 +\sin \phi_1 \sin \phi_2 \cos (\theta_1 - \theta_2 )\right)}. \end{eqnarray*}