Rotations are invertible, their inverse being the rotation by the opposite angle about the same axis.
Observe the action on of both maps. is left invariant by and and is the only subset which has that property.
Thus a necessary condition to the equality is which means or
A similar proof gives: or
Therefore a necessary condition is:
The only thing to show now is that in theese cases, and commute; so the condition will also be sufficient and thus equivalent to the equality.