Inverse Hyperbolic Secant of Imaginary Number

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Theorem

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


Proof

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

$\blacksquare$


Sources