Sine and Cosine are Periodic on Reals/Sine/Proof 1

Theorem
The sine function is periodic with the same period as the cosine function.


 * SineCos.png

Proof
Since Real Cosine Function is Periodic, let $K$ be its period.

Then:
 * $\cos K = \map \cos {0 + K} = \cos 0$

Because Cosine of Zero is One:
 * $\cos K = 1$

Furthermore:

Then, the following holds:

Thus $\sin$ is periodic with some period $L \leq K$.

The following also hold:

So we may conclude:

Therefore the period of cosine is at most that of sine:
 * $K \leq L$

But if $K \leq L \leq K$ then:
 * $K = L$