Is the solution of the equation
$$x + \arctan(x) = \pi$$
The equation of $x + \arctan(x) = 1$ must be transcendental because for any nonzero algebraic $x$, $arctan(x)$ is transcendental, but this argument does not work for the above equation. The continued fraction of the above solution has more than $97000$ terms (PARI), so the answer seems to be yes. But can it be proven ?