.

(*restored equation*)

$$=2 \hbar^2 e^{+2\imath \phi} \left ( \cos^2 \theta- \sin^2 \theta - \cos \theta \cot\theta \sin \theta \right )$$ (52)

which is

$$=2 \hbar^2 e^{+2\imath \phi} \left ( \cos^2 \theta- \sin^2 \theta - \cos^2 \theta \right )$$ (53)

which finally is

$$=-2 \hbar^2 e^{+2\imath \phi} \sin^2 \theta$$ (54)

This is the expected 3d orbital for $m_\ell=+2$ ,i.e., |2,2>.