# Inverse Cosine of Imaginary Number

## Theorem

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

## Proof

 $\displaystyle y$ $=$ $\displaystyle \cos^{-1} x$ $\displaystyle \implies \ \$ $\displaystyle \cos y$ $=$ $\displaystyle x$ Definition of Inverse Cosine $\displaystyle \implies \ \$ $\displaystyle \cosh \paren {i y}$ $=$ $\displaystyle x$ Cosine in terms of Hyperbolic Cosine $\displaystyle \implies \ \$ $\displaystyle i y$ $=$ $\displaystyle \cosh^{-1} x$ Definition of Inverse Hyperbolic Cosine $\displaystyle \implies \ \$ $\displaystyle y$ $=$ $\displaystyle - i \cosh^{-1} x$ multiplying both sides by $-i$

$\blacksquare$