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 $\cos(2\theta)=2\cos^2(\theta)-1$ )

If $x$ is some rational number, what can you say about $2x^2 - 1$?
Using this reasoning, suppose $\cos \theta$ were rational (even though it isn't).  Then what would you know about $\cos 2\theta$?  Why would this be a problem?
Conclude that since $\cos \theta$ being rational results in a problem, $\cos \theta$ must be irrational.
