Complex Cosine Function is Unbounded/Proof 2

Proof
By Complex Cosine Function is Entire, we have that $\cos$ is an entire function.

that $\cos$ was a bounded function.

Then, by Liouville's Theorem, we would have that $\cos$ is a constant function.

However we have, for instance, by Cosine of Zero is One:


 * $\cos 0 = 1$

and by Cosine of 90 Degrees:


 * $\sin \dfrac \pi 2 = 0$

Therefore, $\cos$ is clearly not a constant function, a contradiction.

We hence conclude, by Proof by Contradiction, that $\cos$ is unbounded.