Originally Posted by

**Ackbeet** I think you're forgetting the product rule in a place or two. Also, your $\displaystyle \phi$'s and $\displaystyle \theta$'s are too much alike. Try this:

$\displaystyle \displaystyle-\hbar^{2}e^{i\phi}\left(\frac{\partial}{\partial\t heta}+i\cot(\theta)\,\frac{\partial}{\partial\phi} \right)\left[e^{-i\phi}\left(\frac{\partial f}{\partial\theta}-i\cot(\theta)\,\frac{\partial f}{\partial\phi}\right)\right]$

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

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

$\displaystyle \displaystyle=-\hbar^{2}e^{i\phi}\left[e^{-i\phi}\,\frac{\partial^{2}f}{\partial\theta^{2}}-e^{-i\phi}i\,\underbrace{\frac{\partial}{\partial\thet a}\left(\cot(\theta)\,\frac{\partial f}{\partial\phi}\right)}_{\text{Product Rule}}+i\cot(\theta)\,\underbrace{\frac{\partial}{ \partial\phi}\left(e^{-i\phi}\,\frac{\partial f}{\partial\theta}\right)}_{\text{Product Rule}}+\cot^{2}(\theta)\underbrace{\frac{\partial} {\partial\phi}\left(\,e^{-i\phi}\frac{\partial f}{\partial\phi}\right)}_{\text{Product Rule}}\right].$

See what that does for you.