How do you prove that two languages are equivalent? How can you show that the Language accepted by an NFA and the reverse NFA is the same?

For a language $L$, there is an $L^R=\{ w^R \mid w \in L\}$

Let's say that $w^R$ is the string obtained by reversing the string $w$.

I know that it involves using induction on the length of the input, but I would really appreciate some help.
There's an algorithm for this. Construct a DFA from you NFA, construct the reverse DFA, and check whether the two DFAs accept the same language (for example, you could construct the DFA for the symmetric difference and see if it accepts anything). If you want a more serious answer, you'll need to tell us more about your $L$.