Verify Indifference curves of two utility function to be the same
Here is a way to approach the problem:

An indifference curve is a set of consumption bundles all of which yield the same utility (the consumer being, of course, indifferent to which she consumes). Nonetheless, the numerical level of utility is inconsequential, it is just a placeholder for the more fundamental (and ordinal) preference--any monotone transformation of a utility function represents the preferences. 

So let $I_U(x_1,x_2)$ denote the indifference curve (of $U$) that contains $(x_1,x_2)$. That is $$I_U(x_1,x_2) = \{(x'_1,x'_2) \in X \mid U(x'_1,x'_2) = U(x_1,x_2)\}$$
Define $I_V$ in an analogous way.

So showing the indifference curves are the same amounts to showing $I_U(x_1,x_2)= I_V(x_1,x_2)$ for all $(x_1,x_2) \in X$.