I'm taking a course in Quantum Mechanics and this problem is causing me some struggles. Can someone help me prove this identity?
$$\sum_{m = -l}^l m^2 |Y_{l}^{m}(\theta, \phi)|^2 = \frac{l(l+1)(2l+1)}{8\pi}\sin^2(\theta)$$
I had the idea to use addition theorem for the same $\theta$ and $\phi$ given by $$\sum_{m = -l}^l |Y_{l}^{m}(\theta, \phi)|^2 = \frac{(2l+1)}{4\pi}$$ Also I tried to use some recurrence relation for Legendre polynomials and induction on $l$