I've been struggling with this integral $$ \int_0^{2\pi}\int_0^{\pi} \sin(\theta) e^{-i\phi} Y_{l m}(\theta,\phi) Y^*_{l'm'} (\theta,\phi) \mathrm d\theta \mathrm d\phi $$ I've tried to use the definition of the spherical harmonics as $$ Y_{lm}(\theta,\phi) = N P^m_l(\cos(\theta)) e^{i m \phi} $$ with $N$ being the normalization constant.
I was planning to use the orthogonality of the associated Legendre polynomials $P^n_l$ but I was unable to make any further progress. How do I go around solving the given integral?