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

\(\ds \cosh^2 x - \sinh^2 x\)

\(=\)

\(\ds 1\)

Difference of Squares of Hyperbolic Cosine and Sine

\(\ds \leadsto \ \ \)

\(\ds \cosh^2 x\)

\(=\)

\(\ds 1 + \sinh^2 x\)

\(\ds \)

\(\ge\)

\(\ds 1\)

Square of Real Number is Non-Negative

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

$\blacksquare$