Inverse Secant of Imaginary Number

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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 $\pm \, i$

$\blacksquare$


Sources