How to prove $\cos(x)=\cos(-x)$?

Question

Couldn't I just argue with the definitions $\cos(z):=\frac{1}{2}(e^{iz}+e^{-iz})$?

So $\cos(z):=\frac{1}{2}(e^{iz}+e^{-iz})=\frac{1}{2}(e^{-iz}+e^{iz})=\cos(-z)$

???

Answer 1

Since you are told that the definition of cosine is $$\frac12\left(e^{ix} + e^{-ix} \right),$$

yes, you have proven it by using commutative property. Note that there is a slight inconsitency for confusing $z$ with $x$.

Answer 2

There is nothing wrong with your argument, although some of the notation is wrong, as long as that identity has not been used in the proof that formula, which it has not. There are a plethora of other ways to prove this, though.