Inverse Hyperbolic Cosine of Imaginary Number

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 $-i$

$\blacksquare$