There are some threads that discuss it but I haven't came across an inductive one yet. All of them involve creating a finite automaton which I would like to avoid (as per my professors requests).
You can use regular expressions to easily solve this:
$$(L_1\cup L_2)^r=L_1^r\cup L_2^r$$ $$(L_1\cap L_2)^r=L_1^r\cap L_2^r$$ $$(L_1L_2)^r=L_2^rL_1^r$$ $$(L^*)^r=(L^r)^*$$
You can use these equalities when proving by induction over the number of operators in the regular expression.