How to prove:
$$ \int^\pi_0\int^{2\pi}_0 Y_{l'',m''_l}(\theta,\phi) Y_{l',m'_l}(\theta,\phi) Y_{l,m_l}(\theta,\phi) \sin\theta \,d\theta \,d\phi = 0 $$ unless $l, l',$ and $l''$ are integers that can form the sides of a trangle and $m_l+m_l'+m_l''=0$
$Y_{l,m_l}(\theta,\phi)$ are spherical harmonics
Can I somehow make use of the orthogonality of the spherical harmonics ? i.e, $$ \int^\pi_0\int^{2\pi}_0 Y_{l,m_l}(\theta,\phi) Y_{l',m_l'}(\theta,\phi)\sin\theta \, d\theta \, d\phi=\delta_{l,l'}\delta_{m_l,m_l'} $$
Note: To understand the context, please check here.