Inverse Hyperbolic Cosine of Imaginary Number

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Theorem

$\cosh^{-1} x = \pm \, i \cos^{-1} x$


Proof

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

$\blacksquare$


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