Can you prove that $$ \frac{Y_{\ell}^{m}Y_{\ell'}^{-m}}{\sin^2\theta}=\sum_{L=0}^{\ell+\ell'}a_LY_L^0~, $$ where $Y_{\ell}^{m}$ denotes the spherical harmonics of degree $\ell$ and order $m$ (both are integers)?
The relation seems to hold for all the combinations of the integers $\ell$, $\ell'$ and $m$ that I have tried. But I would like to have an expression for the $a_L$ that works for any $\ell$, $\ell'$ and $m$.