Sine and Cosine are Periodic on Reals/Cosine

Theorem
The cosine function is periodic on the set of real numbers $\R$:
 * $\exists L \in \R_{\neq 0}: \forall x \in \R: \cos x = \map \cos {x + L}$


 * SineCos.png

Proof
From Real Cosine Function has Zeroes, the cosine function has at least one positive zero.

Therefore there exists a Greatest Lower Bound $\eta \in \R_{>0}$ to the set of positive zeroes.

Since Cosine Function is Continuous, $\eta$ is a zero.

Because Cosine Function is Even:
 * $\cos \eta = \map \cos {-\eta} = 0$

By definition of greatest lower bound:
 * $\cos x \neq 0$ for $-\eta < x < \eta$

Because Cosine of Zero is One, it follows from the Intermediate Value Theorem that:
 * $\cos x > 0$ for $-\eta < x < \eta$

From Sum of Squares of Sine and Cosine:
 * $\cos^2 x + \sin^2 x = 1$

Hence as $\cos \eta = 0$ it follows that:
 * $\sin^2 \eta = 1$

So either $\sin \eta = 1$ or $\sin \eta = -1$.

But from Derivative of Sine Function:
 * $\map {\dfrac \d {\d x} } {\sin x} = \cos x$

On the interval $\openint {-\eta} \eta$, it has been shown that $\cos x > 0$.

Thus by Derivative of Monotone Function, $\sin x$ is increasing on $\closedint {-\eta} \eta$.

Since $\sin 0 = 0$ it follows that:
 * $\sin \eta > 0$

So it must be the case that:
 * $\sin \eta = 1$

Now we apply Sine of Sum and Cosine of Sum:

Hence it follows that:

Thus $\cos$ is periodic on $\R$ with a period $L \leq 4 \eta$.