Let $G$ be a group, let $x,u \in G$ show that if $x$ and $y$ commute and have coprime orders then $\langle x,y\rangle$ is cyclic.
Knowing that $$\langle x,y\rangle= \{u_1 u_2, \ldots, u_k: u_i \in \{x,x^{-1}, y,y^{-1}\}\}$$ any element of $\langle x,y\rangle $ can be written as product of $x$s and $y$s powers (i.e $w\in\langle x,y\rangle $ then $w=x^n y^m$ for some natural numbers $n,m$)
Any ideas?
I tried to show that the group is generated by $w=x^n y^m$ but I couldn't reach to anything,