# Help to show this equality of differential operators in spherical coordinates

There are two points on the same sphere with coordinates ${R, \theta_1, \phi_1}$ and ${R, \theta_2, \phi_2}$. Also I have the operator $\displaystyle { \nabla _{{\Omega _1}}^2 + \nabla _{{\Omega _2}}^2}$, where

$$\nabla _{{\Omega _i}}^2 = \frac{1}{\sin \theta_i} \frac{\partial} {\partial \theta_i} \left( \sin \theta_i \frac{\partial}{\partial \theta_i} \right) + \frac{1}{\sin ^ 2 \theta_i} \frac{\partial ^ 2} {\partial \varphi^2 _i}$$

(the angular part of delta operator) and I know that:

$$\cos \vartheta = \frac{(\vec {r_1} \cdot \vec {r_2})}{R^2} = \cos \theta_1 \cos \theta_2 + \sin \theta_1 \sin \theta_2 \cos (\varphi_1 - \varphi_2).$$

I need to prove that

$${ \nabla _{{\Omega _1}}^2 + \nabla _{{\Omega _2}}^2} = 2\left( {\frac{{{d^2}}}{{d{\vartheta ^2}}} + \cot\vartheta \frac{d}{{d\vartheta }}} \right).$$

I assume the operator is understood to act on functions of the form $f(\theta_1,\phi_1,\theta_2,\phi_2)$? – anon Feb 13 '12 at 20:25
@anon yes, sure! and the right side operator acts on $f(\vartheta)$ – Narek Feb 14 '12 at 4:13