Inverse Cotangent of Imaginary Number
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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