Inverse Cotangent of Imaginary Number

Theorem

$\map {\cot^{-1} } {i x} = - i \coth^{-1} x$

Proof

\(\ds y\)

\(=\)

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

\(\ds \leadsto \ \ \)

\(\ds \cot y\)

\(=\)

\(\ds i x\)

Definition of Inverse Cotangent

\(\ds \leadsto \ \ \)

\(\ds i \cot y\)

\(=\)

\(\ds - x\)

$i^2 = -1$

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds x\)

Cotangent in terms of Hyperbolic Cotangent

\(\ds \leadsto \ \ \)

\(\ds i y\)

\(=\)

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

Definition of Inverse Hyperbolic Cotangent

\(\ds \leadsto \ \ \)

\(\ds y\)

\(=\)

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

multiplying both sides by $-i$

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.98$: Relationship between Inverse Hyperbolic and Inverse Trigonometric Functions