Sine and Cosine are Periodic on Reals/Pi/Proof 2

Proof
By Cosine of Zero is One:
 * $\cos 0 = 1$

By Cosine of 2 is Strictly Negative:
 * $\cos 2 < 0$

Thus by the corollary to the Intermediate Value Theorem there exists an $h \in \openint 0 2$ such that:
 * $\cos h = 0$

By Sine of Sum for all $x \in \R$:

By Cosine of Sum for all $x \in \R$:

By Sum of Squares of Sine and Cosine:

Thus for all $x \in \R$:

In particular, $\cos$ is periodic.

By Nonconstant Periodic Function with no Period is Discontinuous Everywhere, $\cos$ has a period $p \in \R_{>0}$.

In view of $(1)$ and $\sin h \ne 0$, the periodic elements of $\sin$ are exactly those of $\cos$.

Thus $p$ is also the period of $\sin$.