Inverse Cosecant of Imaginary Number

Theorem

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

Proof

\(\ds y\)

\(=\)

\(\ds \map {\csc^{-1} } {i x}\)

\(\ds \leadsto \ \ \)

\(\ds \csc y\)

\(=\)

\(\ds i x\)

Definition of Inverse Cosecant

\(\ds \leadsto \ \ \)

\(\ds i \csc y\)

\(=\)

\(\ds -x\)

$i^2 = -1$

\(\ds \leadsto \ \ \)

\(\ds \map \csch {i y}\)

\(=\)

\(\ds -x\)

Cosecant in terms of Hyperbolic Cosecant

\(\ds \leadsto \ \ \)

\(\ds i y\)

\(=\)

\(\ds \map {\csch^{-1} } {-x}\)

Definition of Inverse Hyperbolic Cosecant

\(\ds \leadsto \ \ \)

\(\ds i y\)

\(=\)

\(\ds -\csch^{-1} x\)

Inverse Hyperbolic Cosecant is Odd Function

\(\ds \leadsto \ \ \)

\(\ds y\)

\(=\)

\(\ds i \sinh^{-1} x\)

multiplying both sides by $-i$

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.102$: Relationship between Inverse Hyperbolic and Inverse Trigonometric Functions