While working on finite order elements of $\operatorname{SO}_n$, I meet this question:

Find all identities of the form $\cos(2\pi x)\cos(2\pi y) = \cos(2\pi z)$ with $x, y, z$ rational numbers.

As to background: this is more or less the equation that the product of three finite order elements of $\operatorname{SO}_3(\Bbb R)$ is the identity.

From experimental results, it seems that the only possible solutions are:

• $0 \cdot * = 0$;
• $\pm1 \cdot * = \pm*$;
• $\pm\frac1{\sqrt 2} \cdot \pm\frac1{\sqrt 2} = \pm\frac1 2$.

So, are these the only solutions and how can one prove that?

I abbreviate $\cos(2\pi x)$ to $c(x)$. Using the identity $c(x)c(y) = \frac{c(x - y) + c(x + y)}2$, we can (up to renaming the variables) get an equivalent form of the equation: $c(x) + c(y) = 2c(z)$.

From here I don't find a proper method from my usual arsenal. I tried to look at ramification of the corresponding cyclotomic fields but got nowhere.