Complex Cosine Function is Unbounded/Proof 2
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The complex cosine function is unbounded.
By Complex Cosine Function is Entire, we have that $\cos$ is an entire function.
Aiming for a contradiction, suppose that $\cos$ is a bounded function.
By Liouville's Theorem, we have that $\cos$ is a constant function.
However, by Cosine of Zero is One:
- $\cos 0 = 1$
and by Cosine of Right Angle:
- $\sin \dfrac \pi 2 = 0$
Therefore, $\cos$ is clearly not a constant function.
This contradicts the statement that $\cos$ is a constant function.
We hence conclude, by Proof by Contradiction, that our assumption that $\cos$ is a bounded is false.
Hence $\cos$ is unbounded.