Half Angle Formulas/Hyperbolic Cosine

Theorem

Let $x \in \R$.

Then:

$\cosh \dfrac x 2 = +\sqrt {\dfrac {\cosh x + 1} 2}$

where $\cosh$ denotes hyperbolic cosine.

Proof

 $\displaystyle \cosh x$ $=$ $\displaystyle 2 \ \cosh^2 \frac x 2 - 1$ Double Angle Formula for Hyperbolic Cosine: Corollary 1 $\displaystyle \leadsto \ \$ $\displaystyle 2 \ \cosh^2 \frac x 2$ $=$ $\displaystyle \cosh x + 1$ $\displaystyle \leadsto \ \$ $\displaystyle \cosh \frac x 2$ $=$ $\displaystyle \pm \sqrt {\frac {\cosh x + 1} 2}$

As $\forall x \in \R: \cosh x > 0$, the result follows.

$\blacksquare$