# Inverse Hyperbolic Cosine of Imaginary Number

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## Theorem

- $\cosh^{-1} x = \pm \, i \cos^{-1} x$

## Proof

\(\displaystyle y\) | \(=\) | \(\displaystyle \cosh^{-1} x\) | |||||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \cosh y\) | \(=\) | \(\displaystyle x\) | Definition of Inverse Hyperbolic Cosine | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \cos \paren {i y}\) | \(=\) | \(\displaystyle x\) | Hyperbolic Cosine in terms of Cosine | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle i y\) | \(=\) | \(\displaystyle \cos^{-1} x\) | Definition of Inverse Cosine | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle y\) | \(=\) | \(\displaystyle - i \cos^{-1} x\) | multiplying both sides by $-i$ |

$\blacksquare$

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.95$: Relationship between Inverse Hyperbolic and Inverse Trigonometric Functions