Rational values of $\tan(r\pi)$

Find all the $r \in \mathbb{Q}$ such that $\tan(r\pi) \in \mathbb{Q}$ .

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Solution by UettechatFR

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On admet que les $r \in \mathbb{Q}$ tels que $\cos(r\pi) \in \mathbb{Q}$ est l’ensemble $\frac{1}{3}\mathbb{Z} \cup \frac{1}{2}\mathbb{Z}$ (voir problèmes liés).
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$\cos(2r\pi)=2\cos^{2}(r\pi)-1=\frac{2}{1+\tan^{2}(r\pi)}-1=\frac{1-\tan^{2}(r\pi)}{1+\tan^{2}(r\pi)}$ Si $\tan(r\pi) \in \mathbb{Q}$ , alors $\frac{1-\tan^{2}(r\pi)}{1+\tan^{2}(r\pi)} = \cos(2r\pi) \in \mathbb{Q}$ , donc $r \in \frac{1}{6}\mathbb{Z} \cup \frac{1}{4}\mathbb{Z}$ , donc $\tan(r\pi) \in \{-1,0,1\} \subset \mathbb{Q}$
Donc l’ensemble des solutions est
$r \in \frac{1}{6}\mathbb{Z} \cup \frac{1}{4}\mathbb{Z}$

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