Minimum of Real Hyperbolic Cosine Function

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Theorem

Let $x$ be a real number.

Then:

$\cosh x \ge 1$

where $\cosh$ denotes the hyperbolic cosine function.


Proof

\(\displaystyle \cosh^2 x - \sinh^2 x\) \(=\) \(\displaystyle 1\) Difference of Squares of Hyperbolic Cosine and Sine
\(\displaystyle \implies \ \ \) \(\displaystyle \cosh^2 x\) \(=\) \(\displaystyle 1 + \sinh^2 x\)
\(\displaystyle \) \(\ge\) \(\displaystyle 1\) Square of Real Number is Non-Negative

Furthermore, $\cosh x = 1$ when $x = 0$, satisfying the equality case.

$\blacksquare$