# Inverse Secant of Imaginary Number

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## Theorem

- $\sec^{-1} x = \pm i \sech^{-1} x$

## Proof

\(\displaystyle y\) | \(=\) | \(\displaystyle \sec^{-1} x\) | |||||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \sec y\) | \(=\) | \(\displaystyle x\) | Definition of Inverse Secant | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \sech \paren {i y}\) | \(=\) | \(\displaystyle x\) | Secant in terms of Hyperbolic Secant | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle i y\) | \(=\) | \(\displaystyle \sech^{-1} x\) | Definition of Inverse Hyperbolic Secant | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle y\) | \(=\) | \(\displaystyle -i \sech^{-1} x\) | multiplying both sides by $-i$ |

$\blacksquare$

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.100$: Relationship between Inverse Hyperbolic and Inverse Trigonometric Functions