I've been trying to solve this question from Jon Mathews Mathematical methods for physics, and I'm honestly very lost, I was given the following hint:
$$ \cos \gamma=\cos \theta \cos \theta^{\prime}+\sin \theta \sin \theta^{\prime} \cos \left(\phi-\phi^{\prime}\right) $$
And the problem is problem 7-3:
Consider the integral $$ I=\int d \Omega f(\cos \alpha) g(\cos \beta) $$ where $\alpha$ and $\beta$ are the angles between the variable direction $\Omega$ and two fixed directions in space. The integral may be written in the form $$ I=\int d x d y f(x) g(y) K(x, y) $$ in two ways:
(1) by a change of variable
(2) by expanding $f$ and $g$ in series of Legendre polynomials. By comparing the two expressions, evaluate the sum $$ \sum_{l=0}^{\infty}(2 l+1) P_{l}(x) P_{l}(y) P_{l}(z) $$