Prove that if $\theta$ is an angle with $\cos(2\theta)$ irrational then $\cos \theta$ is also irrational Prove that if $\theta$ is an angle with $\cos(2\theta)$ irrational then $\cos \theta$ is also irrational. (hint: recall that $\cos(2\theta)=2\cos^2(\theta)-1$ )

To add to what people have done here so far, here is a proof that if $x$ is rational, then $x^2$ is rational:

Suppose that $x$ is rational.  Then $x=\frac ab$ for integers $a,b$, with $b\neq 0$.  $a^2$ is an integer, and $b^2$ is a non-zero integer.  Thus, $\frac{a^2}{b^2}=x^2$ is rational.

Now, given that $x=\cos\theta$ is rational, how could you show that $2x^2-1$ is rational?
