I'm trying to calculate this integral:
$$\int_0 ^\pi\int_0^{2\pi} [\sin\theta(\cos\theta \sin \phi + \sin\theta \cos(2\phi))]^2 \sin\theta d\phi d\theta $$
We could compute this integral directly. Or notice that it's a linear combination of product of several spherical harmonics and apply Wigner 3-j symbols.
But I've been told that this is integral is very easy to solve, so there must be an easier method.
Could someone give me an indication.