Inverse Cosecant of Imaginary Number

Theorem

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

$\ds y$

$=$

$\ds \map {\csc^{-1} } {i x}$

$\ds \leadsto \ \ $

$\ds \csc y$

$=$

$\ds i x$

Definition of Inverse Cosecant

$\ds \leadsto \ \ $

$\ds i \csc y$

$=$

$\ds -x$

$i^2 = -1$

$\ds \leadsto \ \ $

$\ds \map \csch {i y}$

$=$

$\ds -x$

$\ds \leadsto \ \ $

$\ds i y$

$=$

$\ds \map {\csch^{-1} } {-x}$

Definition of Inverse Hyperbolic Cosecant

$\ds \leadsto \ \ $

$\ds i y$

$=$

$\ds -\csch^{-1} x$

$\ds \leadsto \ \ $

$\ds y$

$=$

$\ds i \sinh^{-1} x$

multiplying both sides by $-i$

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: Relationship between Inverse Hyperbolic and Inverse Trigonometric Functions: $8.102$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 14$: Hyperbolic Functions: Relationship between Inverse Hyperbolic and Inverse Trigonometric Functions: $14.102.$