Proving equivalency of regular expressions
Conceptually, the simplest approach is:


Convert both regular expressions to NFA.
Determinise both NFA.
Minimize both DFA.
The regular expressions are equivalent if and only if the minimal DFA are isomorphic.


The requisite algorithms and theorems are taught in (about) every introducory class to formal languages resp. automata and can be found in (every) textbook on the topic. In fact, the point of this exercise probably was to come up with this approach, combining results from the course. Sorry for spoiling it.