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Quantum Mechanics - Angular Momentum (Eigen Functions) part 4

Hi Folks,

I am having difficulty expanding this exercise.

$\displaystyle

\displaystyle L_+ L_- f = - \hbar ^{2} e^{i \phi} \left (\frac{\partial}{\partial \theta}+i cot \theta \frac{\partial}{\partial \phi} \right ) \left [e^{-i \phi} \left (\frac{\partial f}{\partial \theta}-i cot \theta \frac{\partial}{\partial \phi} \right )\right ]

$

My attempt involved as attached but the next line of derivation has 3 more terms...I thought it was just a matter of multiplying out terms and simplifying...?

The answer is

$\displaystyle

\displaystyle L_+ L_= - \hbar ^{2}\left (\frac{\partial^{2}}{\partial \theta^{2}}+cot \theta \frac{\partial}{\partial \theta} + cotan^{2} \theta \frac{\partial^{2}}{\partial \phi^{2}} +i \frac{\partial}{\partial \phi}\right )

$

Thanks