Inverse Hyperbolic Secant of Imaginary Number

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Theorem

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


Proof

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

$\blacksquare$


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