Half Angle Formulas/Hyperbolic Cosine

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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.

$\blacksquare$


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