While solving a different problem, I found Lambert’s proof (and later Laczkovich’s simplification of Lambert’s proof) that $\pi$ is irrational:
https://en.m.wikipedia.org/wiki/Proof_that_π_is_irrational#Laczkovich.27s_proof
But then I found this question on SE
That asks for a proof for the fact that given a nonzero rational $x$, show that $\tan(x)$ is irrational (ironically, in order to later show that $\pi$ is irrational).
This got me thinking about a problem I’ve now been stuck on for a bit. If I want to generalize this claim a little more, then what can we say about $\tan$ with respect to $\mathbb{Q}(\sqrt{2})$. Is it true that for $x \in \mathbb{Q}(\sqrt{2})$ we have $\tan(x) \in \mathbb{R} - \mathbb{Q}(\sqrt{2})$? If not that when can we say that $\tan(x) \in \mathbb{R} - \mathbb{Q}(\sqrt{2})$?
Edit: where $x$ is nonzero of course
