Is $\tan x\in\mathbb{Q}\setminus\{0\}\implies x\notin\mathbb{Q}$ always true?

Question

While proving that $\pi$ is irrational, Johann Heinrich Lambert proved that $$x\in\mathbb{Q}\setminus\{0\}\implies \tan x\notin\mathbb{Q},$$ but is it always true that $$\tan x\in\mathbb{Q}\setminus\{0\}\implies x\notin\mathbb{Q}?$$ If not, what is $x$ such that the statement is not true?

Answer 1

The contrapositive of the statement is that if $\tan x\in\mathbb Q$, then $x\notin\mathbb Q-\{0\}$. Thus if $\tan x\in \mathbb Q-\{0\}$, then it's still true that $x\notin \mathbb Q-\{0\}$, but additionally $x\neq 0$ because $\tan x\neq 0$, so $x\notin\mathbb Q$.