Secant of i
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Theorem
- $\sec i = \dfrac {2 e} {e^2 + 1}$
where $\sec$ denotes the complex secant function and $i$ is the imaginary unit.
Proof 1
| \(\ds \sec i\) | \(=\) | \(\ds \frac 1 {\cos i}\) | Definition of Complex Secant Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {\frac e 2 + \frac 1 {2 e} }\) | Cosine of $i$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {2 e} {e^2 + 1}\) | multiplying denominator and numerator by $2 e$ |
$\blacksquare$
Proof 2
| \(\ds \sec i\) | \(=\) | \(\ds \sech 1\) | Hyperbolic Secant in terms of Secant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 2 {e^1 + e^{-1} }\) | Definition of Hyperbolic Secant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {2 e} {e^2 + 1}\) | multiplying denominator and numerator by $e$ |
$\blacksquare$