I'm asked to express the function $$ f(\theta,\phi) = \sin\theta \left[ \sin^2\frac{\theta}{2}\,\cos\phi + i \cos^2\frac{\theta}{2}\,\sin\phi \right] + \sin^2\frac{\theta}{2}$$ as a sum of spherical-harmonics $Y_l^m(\theta,\phi)$.
So, I have to write $f$ in the form $$ f(\theta,\phi) = \sum_{m,l} a_{ml}Y_l^m(\theta,\phi). $$ I'm not clear on how to get the values for $a_{ml}$. I was thinking on evaluating the integral $$ a_{ml} = \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi} Y_l^{m*}(\theta,\phi)\sin\theta \,f(\theta,\phi) d\theta d\phi, $$ where $f(\theta,\phi)$ is written as the first equation, but I'm unable to solve this integral. Ideas?
Also, having $\sin^2$ and $\cos^2$ means that $l=0,1,2$ or just $l=2$ ?