There are infinite products of iterated square roots for $\log x$ and $\arccos x$ as functions of $x$. For example

$$\log x = \frac{x - 1}{\sqrt{x}\sqrt{\frac{1}{2} + \frac{1}{2}\left ( \frac{1 + x}{2\sqrt{2}} \right )}\sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2} + \frac{1}{2}\left ( \frac{1 + x}{2\sqrt{2}} \right )\cdots}}} \tag{1}$$

$$\frac{\sqrt{1-x^2}}{\arccos x} = \frac{\sqrt{2+2x}}{2}\frac{\sqrt{2+\sqrt{2+2x}}}{2}\frac{\sqrt{2+\sqrt{2+\sqrt{2+2x}}}}{2}\cdots\tag{2}$$

Keeping this in mind, I wonder if there is an infinite product of square roots, similar to those products above, for $\cos x$, as a function of $x$.

Are the identities $(1)$ and $(2)$ helpful in obtaining a similar product for $ \cos x $? If not, is there any infinite product with square roots, which can be derived in some way, or is there some demonstration that such a product can not exist?

References:

*An infinite product of nested radicals for from
the Archimedean algorithm log x*

Thomas J. Osler, Walter Jacob and Ryo Nishimura.