# 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.

$\blacksquare$