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

\(\ds \cosh x\)

\(=\)

\(\ds 2 \cosh^2 \frac x 2 - 1\)

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

• Half Angle Formula for Hyperbolic Sine
• Half Angle Formula for Hyperbolic Tangent

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.28$: Double Angle Formulas