# Inverse Hyperbolic Cosine of Imaginary Number

## Theorem

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

## Proof

 $\ds y$ $=$ $\ds \cosh^{-1} x$ $\ds \leadsto \ \$ $\ds \cosh y$ $=$ $\ds x$ Definition of Inverse Hyperbolic Cosine $\ds \leadsto \ \$ $\ds \map \cosh {\pm \, y}$ $=$ $\ds x$ Hyperbolic Cosine Function is Even $\ds \leadsto \ \$ $\ds \map \cos {\pm \, i y}$ $=$ $\ds x$ Hyperbolic Cosine in terms of Cosine $\ds \leadsto \ \$ $\ds \pm \, i y$ $=$ $\ds \cos^{-1} x$ Definition of Inverse Cosine $\ds \leadsto \ \$ $\ds y$ $=$ $\ds \pm \, i \cos^{-1} x$ multiplying both sides by $\pm \, i$

$\blacksquare$