Solve for $f(x)$ if $f(x)\cdot f(-x)=g(x)$.
I have been having trouble figuring this out. I asked ChatGPT, and its answers don't work. I then looked online (I googled it), and was surprised to find nothing about it.
I was playing around in Desmos and found that you can express sin(x) and cos(x) by using factorials, like so:
$$ \left(-\frac{x+\frac{\pi}{2}}{\frac{x+\frac{\pi}{2}}{\pi}!\cdot-\left(-\frac{x+\frac{\pi}{2}}{\pi}\right)!},-\frac{x}{\frac{x}{\pi}!\cdot-\left(-\frac{x}{\pi}\right)!}\right) $$
$\cos(x)$ can be derived by adding $\frac\pi2$ to every instance of $x$.
I was trying to figure out if it is possible to express $x!$ in terms of $\sin(x)$ (without using any factorials or series (plural) or sums or whatever beyond that of the definition of the sine function itself), and got as far as the following:
$$ \pi x\frac{1}{\sin\left(\pi x\right)}=x!\cdot\left(-x\right)! $$
It can be simplified to $\pi x\csc\left(\pi x\right)$.
Given that $f\left(x\right)=x!$ and $g\left(x\right)=\pi x\csc\left(\pi x\right)$, I want to find $f(x)$ in terms of $g(x)$.
Even better if the problem can be solved regardless of equation. Thanks!
Edit: I also asked Wolfram Alpha, and it worked up until $f(x)\cdot f(x)=g(x)$, but not for $f(x)\cdot f(-x)=g(x)$. Also, if the problem is impossible, I would like to know why. Thanks again!