MathNet 2007-part-03 — 185 problems (EN–VI) Question 167

Let $\theta$ be an angle in the interval $(0, \pi/2)$. Given that $\cos \theta$ is irrational, and that $\cos k\theta$ and $\cos[(k+1)\theta]$ are both rational for some positive integer $k$, show that $\theta = \pi/6$.