Suppose $a \in \langle b\rangle$. Then $\langle a\rangle = \langle b\rangle$ if and only if $a$ and $b$ have the same order.
Here is what I have so far.
Suppose that $a$ has order $n$. If $a \in \langle b\rangle$ then $a=b^k$ for some $k \in \mathbb{Z}$.
$\langle a\rangle = \langle b^k\rangle = \{e, b^k, b^{2k}, \dots, b^{k(n-1)}\}$
I don't really know where to take it from here. Any help would be appreciated!
I guess I'm very confused about when two cyclic subgroups are equal and what does that mean exactly. Does that mean they're generated by the same element?