Differential Equation in Spherical Harmonics Derivation

Question

I've been reviewing derivations of the spherical harmonics in quantum mechanics; mostly as review but also to make sure I understand where the concepts arise from.

However, every derivation I've seen makes use of the following differential equation/ identity that seemingly comes from nowhere. I've not yet been able to figure out where it comes from in looking through various texts. So, here I am to ask: where does this come from?

$$\frac{d}{d\theta}+l cot(\theta) \equiv \frac{1}{(sin(\theta))^l} \frac{d}{d\theta}(sin(\theta)^l)$$

Answer 1

It comes from various differentiation rules. It is probably easier to grasp if one applies both sides of the identity on some function $f(\theta)$. Then the identity becomes

$$\frac{1}{\sin^\ell\theta} \frac{d}{d\theta}(\sin^\ell\theta~ f(\theta))~=~ f^\prime(\theta)+\ell \cot\theta ~f(\theta). $$