The following infinite product is well known:
$ (1) \frac{\sqrt{1-\alpha^2}}{\arccos \alpha} = \frac{\sqrt{2+2\alpha}}{2}\frac{\sqrt{2+\sqrt{2+2\alpha}}}{2}\frac{\sqrt{2+\sqrt{2+\sqrt{2+2\alpha}}}}{2}...$
For $\alpha \neq 1$
This gives an infinite product for $\arccos \alpha$ similar to Viete's formula for $\pi$.
The question is the following: is there any way to get a similar infinite product for $\cos \alpha$ in terms of $\alpha$ and nested square roots by using some process of inversion applied to formula $(1)$ for $\arccos \alpha$?
If yes, is it always possible to derive a similar representation (infinite series, infinite product, continued fraction, etc.) for a function or its inverse, if we have a representation of one of them?