Inverse Secant of Imaginary Number

Theorem

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

Proof

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

$\blacksquare$