**Example 1**: In some textbooks, the trigonometric functions are defined via geometry. The advantage is that it is simpler to understand, but the disadvantage is that it is difficult to make completely rigorous without using integrals or at least a notion of area equivalent to the Jordan measure. In contrast, it is relatively easy to define them via their power series in a completely rigorous manner, and then prove their desired properties. Moreover, the first approach is restricted to $ℝ$, and we need to do something later to extend to $ℂ$, whereas in the second approach we can get the complex trigonometric functions directly. Nevertheless, both definitions (if done right) are equivalent.

**Example 2**: The Lebesgue measure has a number of equivalent definitions, as mentioned in Terence Tao's "[An Introduction to Measure Theory](https://terrytao.wordpress.com/books/an-introduction-to-measure-theory/)". One definition is that a set $S⊆ℝ^n$ is measurable iff for every $ε>0$ there is some open set $G⊇S$ such that there is a countable sequence of boxes whose union contains $G∖S$ and whose total volume is at most $ε$. Another definition is the [Caratheodory criterion](https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_criterion); $S$ is measurable iff for every $A⊆ℝ^n$ we have $m^*(A) = m^*(A∩S)+m^*(A∖S)$, where $m^*$ is the outer measure. Although the Caratheodory criterion is unintuitive (as Tao also says), there are some textbooks that *define* the Lebesgue measure using that and then *prove* all the other equivalents.