I start with the following:
$\frac{d^n}{dx^n} \left[(1-x^2)^{n+\alpha-1/2}\right]$
Which is part of the Rodrigues definition of a Gegenbauer polynomial. Gegenbauer polynomials are also useful in terms of trigonometric functions so I want to use the substitution $x = \cos\theta$, which is the usual way of doing it. However, I'm stuck as to how this works for the Rodrigues definition, because it gives me a derivative with respect to $\cos\theta$ instead of a derivative with respect to $\theta$:
$\frac{d^n}{d(\cos\theta)^n} \left[(\sin^2\theta)^{n+\alpha-1/2}\right]$
QUESTION: Is there a way to write this as $\dfrac{d^n}{d\theta^n}[\text{something}]$?
I have read some about Faa di Bruno's formula for the $n$-th order derivative of a composition of functions but it doesn't seem to do what I want to do.
Also, for n=1 there is the identity, from the chain rule, $\dfrac{d}{d(\cos\theta)} \left[(\sin^2\theta)^{n+\alpha-1/2}\right]=\frac{\frac{d}{d\theta} \left[(\sin^2\theta)^{n+\alpha-1/2}\right]}{\frac{d}{d\theta} \left[\cos\theta\right]}$, but this doesn't hold for higher order derivatives. Any ideas?