Cosine of Integer Multiple of Pi

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

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

Then:
 * $\cos n\pi = (-1)^n$ for all $n \in \Z$

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 Cosine of Zero is One, we have that:
 * $\displaystyle \cos 0 = 1 - \frac {0^2} {2!} + \frac {0^4} {4!} - \cdots = 1$

This fact is sufficient for the derivation in Sine and Cosine are Periodic on Reals.

Therefore, by the corollary that $\cos(x + \pi) = -\cos x$ and induction we have:
 * $\cos n \pi = \left({-1}\right)^n$