Spherical Harmonic Decomposition

The $\cos \theta$ term comes purely from $Y_1^0$ . To obtain the $\sin^2 \theta \cos (2\phi)$ term, note that the $Y_2^{\pm 2}$ spherical harmonics both include $\sin^2 \theta e^{\pm 2i \phi}$ , and therefore using linear combinations of them one may obtain this term. These three are therefore sufficient.

With little work one can verify that $f(\theta, \phi) = \cos\theta - \sin^{2}\theta \cos(2\phi) = \sqrt{ \frac{4\pi}{3} } \left[ Y_{0}^{1} - \sqrt{ \frac{8}{20} } (Y_{2}^{2} + Y_{2}^{-2}) \right]$