# Inverse Secant of Imaginary Number

## Theorem

$\sec^{-1} x = \pm i \sech^{-1} x$

## Proof

 $\displaystyle y$ $=$ $\displaystyle \sec^{-1} x$ $\displaystyle \leadsto \ \$ $\displaystyle \sec y$ $=$ $\displaystyle x$ Definition of Inverse Secant $\displaystyle \leadsto \ \$ $\displaystyle \map \sec {\pm \, y}$ $=$ $\displaystyle x$ Secant Function is Even $\displaystyle \leadsto \ \$ $\displaystyle \map \sech {\pm \, i y}$ $=$ $\displaystyle x$ Secant in terms of Hyperbolic Secant $\displaystyle \leadsto \ \$ $\displaystyle \pm \, i y$ $=$ $\displaystyle \sech^{-1} x$ Definition of Inverse Hyperbolic Secant $\displaystyle \leadsto \ \$ $\displaystyle y$ $=$ $\displaystyle \pm \, i \sech^{-1} x$ multiplying both sides by $\pm \, i$

$\blacksquare$