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.