Inverse Hyperbolic Cosecant of Imaginary Number

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Theorem

$\map {\csch^{-1} } {i x} = -i \csc^{-1} x$


Proof

\(\ds y\) \(=\) \(\ds \map {\csch^{-1} } {i x}\)
\(\ds \leadsto \ \ \) \(\ds \csch y\) \(=\) \(\ds i x\) Definition of Inverse Hyperbolic Cosecant
\(\ds \leadsto \ \ \) \(\ds i \csch y\) \(=\) \(\ds - x\) $i^2 = -1$
\(\ds \leadsto \ \ \) \(\ds \map \csc {i y}\) \(=\) \(\ds x\) Hyperbolic Cosecant in terms of Cosecant
\(\ds \leadsto \ \ \) \(\ds i y\) \(=\) \(\ds \csc^{-1} x\) Definition of Inverse Cosecant
\(\ds \leadsto \ \ \) \(\ds y\) \(=\) \(\ds -i \csc^{-1} x\) multiplying both sides by $-i$

$\blacksquare$


Sources