China Western Mathematical Olympiad

First we prove a lemma.
Lemma Let $\alpha$ be a real number. If $\cos \alpha$ is rational, then $\cos m\alpha$ is rational for any positive integer $m$.
We prove by induction on $m$. By $\cos 2\alpha = 2\cos^2\alpha - 1$, we get that the lemma is true for $m = 2$.
We suppose that the lemma is true for $m \le l$ ($l \ge 2$).
Since
$$\cos(l + 1)\alpha = 2\cos l\alpha \cdot \cos \alpha - \cos(l - 1)\alpha,$$
then we conclude that the lemma is true for $m = l + 1$ and our induction is complete.
By the lemma, setting $m = k$, $m = k + 1$ for $\alpha = k\theta$, $(k-1)\theta$, it follows that $\cos k^2 \theta$, $\cos(k^2 - 1)\theta$ are rational numbers. Since $k^2 > k$, the statement holds.