When we want to prove that a property P(n) holds for every natural number n, we can, and must use mathematical induction. So I was wondering if it is wrong if we DON'T use induction in obvious mathematical statements. For example, let's solve this exercise below, without mathematical induction.
Exercise: Prove that $n^2-1=(n-1)(n+1)$ for every $n\in \mathbb{N}.$
Solution: Suppose $n\in \mathbb{N}$. Then $(n-1)(n+1)=n\cdot n + n\cdot 1 -1 \cdot n - 1\cdot 1=n^2-1$.
Since $n\in \mathbb{N}$ was arbitary, $$n^2-1=(n-1)(n+1)$$ holds for every $n\in \mathbb{N}.$
So is the above solution correct? Are we obliged to use only mathematical induction to prove that such a statement holds for every $n \in \mathbb{N}$?
You don't HAVE to use induction when working on natural numbers. You MAY use it, and you SHOULD at least give it a try if nothing obvious appears. It's a really powerfull tool, but some cases, just like the example in your question can be solved directly without having to rely on induction