The equation $x=\cos x$ is well-known because some facts. For example, with an old calculator, you can find approximations of the solution by typing any number and pressing the $\cos$ button repeatedly.
With Bolzano-Rolle combo, it is not difficult to show that this solution exists and is unique.
My question: Is there some work about the rationality or trescendality of this solution? Of course, $x$ is in radians.
If the solution is algebraic, then both $x$ and $\cos x$ are algebraic. This implies $\sin x=\pm\sqrt{1-\cos^2x}$ is also algebraic. So $e^{ix}=\cos x+i\sin x$ is also algebraic, which is not possible for any non-zero algebraic $x$ by Lindemann-Weierstrass Theorem.
Hence the solution is transcendental.