Try

```
sage: F.<x,y> = FreeGroup()
sage: G = F / [x^2, y^3, (x*y)^5]
sage: xx = G(x)
sage: yy = G(y)
sage: (xx*yy)^5 == G.one()
True
sage: (xx*yy) == G.one()
False
```

Now knowing that (x*y) is not the identity in the quotient and that its order divides 5 you have no choice.

note: your group G is isomorphic to A5 by sending x to (1,2)(3,4) and y to (1,3,5).

*edit:*
To actually get the coset one can use the functionalities of GAP

```
sage: U = G.subgroup([G(x*y)])
sage: for rc in G.gap().RightCosets(U.gap()): print rc
RightCoset(Group( [ x*y ] ),<identity ...>)
RightCoset(Group( [ x*y ] ),x)
RightCoset(Group( [ x*y ] ),y)
RightCoset(Group( [ x*y ] ),y*x)
RightCoset(Group( [ x*y ] ),y*x*y)
RightCoset(Group( [ x*y ] ),(y*x)^2)
RightCoset(Group( [ x*y ] ),y*x*y^2)
RightCoset(Group( [ x*y ] ),(y*x)^2*y)
RightCoset(Group( [ x*y ] ),(y*x)^2*y^2)
RightCoset(Group( [ x*y ] ),(y*x)^2*y^2*x)
RightCoset(Group( [ x*y ] ),y*x*(y*x*y)^2)
RightCoset(Group( [ x*y ] ),y*x*(y*x*y)^2*x)
```