The question that follows is a continuation of this Stage $1$ question and this previous Stage $2$ question which are needed as part of a derivation of the Associated Legendre Functions Normalization Formula: $$\color{blue}{\displaystyle\int_{x=-1}^{1}[{P_{L}}^m(x)]^2\,\mathrm{d}x=\left(\frac{2}{2L+1}\right)\frac{(L+m)!}{(L-m)!}}$$ where for each $m$, the functions $${P_L}^m(x)=\frac{1}{2^LL!}\left(1-x^2\right)^{m/2}\frac{\mathrm{d}^{L+m}}{\mathrm{d}x^{L+m}}\left(x^2-1\right)^L\tag{1}$$ are a set of Associated Legendre functions on $[−1, 1]$.
The question in my textbook asks me to
Use $$\begin{align}\frac{\mathrm{d}^{L-m}}{\mathrm{d}x^{L-m}}\left(x^2-1\right)^L&=\frac{(L-m)!}{(L+m)!}\left(x^2-1\right)^m\frac{\mathrm{d}^{L+m}}{\mathrm{d}x^{L+m}}\left(x^2-1\right)^L\end{align}\quad \longleftarrow\text{(Stage 1)}$$ to show that $${P_L}^{m}(x)=(-1)^m\frac{(L+m)!}{(L-m)!}\frac{1}{2^LL!}\left(1-x^2\right)^{-m/2}\frac{\mathrm{d}^{L-m}}{\mathrm{d}x^{L-m}}\left(x^2-1\right)^L\tag{2}$$
Start of attempt:
I started with Stage $2$: $${P_L}^{-m}(x)=(-1)^m\frac{(L-m)!}{(L+m)!}{P_L}^{m}(x)\quad \longleftarrow\text{(Stage 2)}$$ and rearranged it to get
$$\begin{align}\require{enclose}{P_L}^{m}(x)&= \frac{1}{(-1)^m}\frac{(L+m)!}{(L-m)!}{P_L}^{-m}\\&= (-1)^m\frac{(L+m)!}{(L-m)!}\frac{1}{2^LL!}\left(1-x^2\right)^{-m/2}\frac{\mathrm{d}^{L-m}}{\mathrm{d}x^{L-m}}\left(x^2-1\right)^L\quad\longleftarrow\bbox[#F80]{\text{Using (1) with -m}}\end{align}$$ as required.
End of attempt.
So I have proved formula $(2)$; but I did not use the formula which it asked me to use.
So the question I have is: Did I really answer the question? Or is there some way it can be answered by using that formula?
For additional context please see the page below which is the source of questions from my textbook:
If you have read the above extract and believe Problem $9$ cannot be done using Problem $7$ alone then please state this in a comment or an answer as there could be a typo in the book. Personally I think Problem $9$ should read "Use Problem $\color{#180}{8}$ to show that ....". If you agree with me please say something; I just need some kind of feedback.
Many thanks.
First of all:
But, since you have proven formula (9) by means of (8) and you have also proven formula (8) by means of (7) you have mastered exercise (9) in all its particulars.
Please note, that use formula (7) does not imply use formula (7) alone. You are free to find your own derivation as long as formula (7) is an essential aspect in your proof.