Is $\tan^{-1}\tan^{-1}1$ irrational?

Question

• Here, it is proven that $\arctan(2)$ is irrational.

• Here, it is proven that $\arctan(x)$ is irrational for natural $x$. By a proof similar to that from the last linked post, it can easily be shown that $\arctan \frac 1x$ is irrational for natural $x$.

• Here, it is proven that $\arctan(x)$ is a rational multiple of $\pi$ iff $(1+xi)^n$ is a real number for some positive integer $n$.

With these in mind,

I am wondering if $\tan^{-1}(\tan^{-1}(1))$ is irrational.

It probably is, but I have yet to prove it. We can write $\tan^{-1}1$ as $\frac{\pi}4$ which follows from the fact that $(1+i)^4 = -4$, but I am not sure how to use this information further. I suspect the proof of this is unreachable, though MSE has surprised me in the past. With this in mind, I have a few related questions, in order of how unlikely they are to be answered:

1. Is $\tan^{-1}\tan^{-1}1$ transcendental?
2. Is $\tan^{-1}\tan^{-1}1$ irrational?
3. Is there any literature on whether $\tan^{-1}\tan^{-1}1$ or a related evaluation of $\arctan$ is irrational/transcendental?
4. Are there any open conjectures which, if true, the irrationality/transcendentality of $\tan^{-1}\tan^{-1}1$ would follow?

Answer 1

1. Almost certainly yes.

2. Almost certainly yes.

3. I doubt it.

4. Schanuel's conjecture, I think.