Group Theory about proving there exists m,n such that $(x ∗ y)^{−1}= y^m ∗ x^n$

Question

I have been trying to solve this question from a past exam for a very long time but don't know how to go about it. I tried making the LHS of the equation equal the identity but that didn't go far.

Any help is appreciated.

Let $(G, ∗, I)$ be a group. Let $x, y ∈ G$ both have finite order. Prove that there exist $m, n ≥ 0$ such that $(x ∗ y)^{−1}= y^m ∗ x^n$.

Answer 1

Since $x,y$ is of finite order, there exists some positive integer $k,l$ such that $x^k=y^l=id$. Now note that $(x*y)^{-1}=y^{-1}*x^{-1}=y^{l-1}*x^{k-1}$.