Cosine Function is Even

Theorem
Let $x \in \R$ be a real number.

Let $\cos x$ be the cosine of $x$.

Then:
 * $\cos \left({-x}\right) = \cos x$

That is, the cosine function is even.

Proof
Recall the definition of the cosine function:


 * $\displaystyle \cos x = \sum_{n=0}^\infty \left({-1}\right)^n \frac {x^{2n}}{\left({2n}\right)!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$

From Even Powers are Positive, we have that:
 * $\forall n \in \N: x^{2n} = \left({-x}\right)^{2n}$

The result follows.