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
\(\ds \cosh x\) | \(=\) | \(\ds 2 \cosh^2 \frac x 2 - 1\) | Double Angle Formula for Hyperbolic Cosine: Corollary $1$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \cosh^2 \frac x 2\) | \(=\) | \(\ds \cosh x + 1\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cosh \frac x 2\) | \(=\) | \(\ds \pm \sqrt {\frac {\cosh x + 1} 2}\) |
As $\forall x \in \R: \cosh x > 0$, the result follows.
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.28$: Double Angle Formulas