# Integrate orthogonal function over solid angle

How do I integrate a product of Legendre polynomials over a volume?

So I understand that bunch of complete basis orthogonal basis as well. i.e. for Legendre polynomial, $$\int_{-1}^1P_n(x)P_m(x)dx=\delta_m^n\frac{2}{2n+1}$$. However, I'm not sure how this was applied/transferred into the integration of solid angle. $$\int_{4\pi} P_n(cos(\theta)) P_m(cos(\theta)) d\Omega$$.

Can you provide a proof please?

In general, how does integration over sold angle work for the usual orthogonal basis? Especially, in ($$sin,cos$$ basis), (Hermite polynomials basis), and (Spherical harmonics basis), are there another form that followed the orthogonal condition as well?