Show that the spherical harmonic function Y_{11}(\theta, \phi) satisfies the angular Equation (7.11).
\frac{1}{\sin \theta} \frac{d}{d \theta}\left(\sin \theta \frac{d f}{d \theta}\right)+\left[\ell(\ell+1)-\frac{m_{\ell}{ }^{2}}{\sin ^{2} \theta}\right] f=0 (7.11)
Strategy We insert the value for Y_{11}(\theta, \phi) into Equation (7.11) with \ell=1 \text { and } m_{\ell}=1 \text {. Because } Y(\theta, \phi)=f(\theta) g(\phi) [see Equation (7.16)], and \theta \text { and } \phi are independent variables, we will be able to separate the constants and variable \phi from the factors involving \theta .
Y(\theta, \phi)=f(\theta) g(\phi) (7.16)