# Minimum of Real Hyperbolic Cosine Function

## 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.

$\blacksquare$