The exercise is to show that if $n = 3k$, then the group presentation $\langle x, y \mid x^n = y^2 = 1,\ xy = yx^2 \rangle$ generates $D_6$. The group presentation for $D_6$ given in the book is $D_6 = \langle r, s \mid r^3 = y^2 = 1,\ rs = sr^{-1} \rangle$.
Now, my aim was to show first that $x^3 = 1$ and then that $x$ and $x^2$ are distinct. However, I'm having issues showing that $x \neq 1$, or equivalently that $x$ and $x^2$ are distinct elements. I know I somehow must use that $n = 3k$ since I didn't need it to show $x^3 = 1$. What confuses me is that I can set $x = y = 1$ and all the relations given in the original presentation would be satisfied.