---

Statement

If $\theta/\pi\in\mathbb Q$, $\tan\theta$ is defined, and $\tan^2\theta\in\mathbb Q$, then $\tan^2\theta\in\{0,1/3,1,3\}$.

Source and scope. Beeson–Laczkovich–Zhang, arXiv:2604.03609v3, Lemma 24, p. 12. Complete rewritten deduction from the following external result, stated as Lemma 23 in the source: if $\theta/\pi$ and $\cos\theta$ are rational, then $\cos\theta\in\{0,\pm1/2,\pm1\}$ (I. Niven, Irrational Numbers, 1967, Corollary 3.12, p. 41).

Proof

The identity $\cos(2\theta)=(1-\tan^2\theta)/(1+\tan^2\theta)$ makes $\cos(2\theta)$ rational. Apply the stated cosine theorem to $2\theta$. The value $-1$ is impossible when $\tan\theta$ is defined. For the four remaining values, solve $\tan^2\theta=(1-\cos(2\theta))/(1+\cos(2\theta))$ to obtain $1,1/3,3,0$, respectively.

Bears on. Problem 633.