Properties of Real Cosine Function

Theorem
The cosine function satisfies the following properties:.


 * 1) $\cos x$ is continuous on $\R$.
 * 2) $\cos x$ is absolutely convergent for all $x \in \R$.
 * 3) $\cos n\pi = (-1)^n$ for all $n \in \Z$
 * 4) $\cos x$ is even, that is, $\cos \left({-x}\right) = \cos x$ for all $x \in \R$.

Proof
Recall the definition:


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

1. Continuity of $\cos x$:

2. Absolute convergence of $\cos x$:

For


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

to be absolutely convergent we want


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

to be convergent.

But


 * $\displaystyle \sum_{n=0}^\infty \frac {\left|{x}\right|^{2n}}{\left({2n}\right)!}$

is just the terms of


 * $\displaystyle \sum_{n=0}^\infty \frac {\left|{x}\right|^n}{n!}$ for even $n$.

Thus


 * $\displaystyle \sum_{n=0}^\infty \frac {\left|{x}\right|^{2n}}{\left({2n}\right)!} < \sum_{n=0}^\infty \frac {\left|{x}\right|^n}{n!}$.

But


 * $\displaystyle \sum_{n=0}^\infty \frac {\left|{x}\right|^n}{n!} = \exp \left|{x}\right|$

from the Taylor Series Expansion for Exponential Function of $\left|{x}\right|$, which converges for all $x \in \R$.

The result follows from the Squeeze Theorem.

3. $\cos n\pi = (-1)^n$:

Follows directly from the definition: $\displaystyle \cos 0 = 1 - \frac {0^2} {2!} + \frac {0^4} {4!} - \cdots = 1$.

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 $\cos n\pi = (-1)^n$

4. $\cos \left({-x}\right) = \cos x$:

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

The result follows from the definition.