Let $A,B\in $ SO$(3)$ so that $A, B$ are rotations by an angle $\theta\in \mathbb R\setminus \mathbb Q$ about the $z$-axis and $x$-axis, respectively. I want to prove that the group $\langle A, B\rangle$ is isomorphic to $F_2$, the free group on $2$ generators.
It suffices to show that no word in the letters $A, A^{-1}, B, B^{-1}$ has a nontrivial relation (i.e., other than something like $AA^{-1}=e)$. And in principle an induction argument on the number of letters should work for this, but I have not been able to produce a valid one.
Could someone please help with the proof?