So the point of introducing this odd-looking representation of the lowering operator is that the \(\sin^{1-l}\theta\) term in the middle is exactly canceled when the operator is applies twice, and sim...So the point of introducing this odd-looking representation of the lowering operator is that the \(\sin^{1-l}\theta\) term in the middle is exactly canceled when the operator is applies twice, and similar cancellations occur on repeating the operation, giving the (relatively) simple representation: \[ Y^m_l(\theta,\phi)=c_l\sqrt{\dfrac{(l+m)!}{(2l)!(l-m)!}}e^{im\phi}\sin^{-m}\theta\dfrac{d^{l-m}}{d(\cos\theta)^{l-m}}\sin^{2l}\theta \label{4.4.24}\]
