Question

prove that some examples 1/AB =1/B.1/A​

Answer 1

Answer:

Your way is absolutely fine. As you note, there is in fact an easier way. It would be enough to show that the element c such that (ab)c=e is in fact c=b−1a−1:

(ab)b−1a−1=a(bb−1)a−1=aea−1=aa−1=e.