Cosecant of i
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Theorem
- $\csc i = \paren {\dfrac {2 e} {1 - e^2} } i$
where $\csc$ denotes the complex cosecant function and $i$ is the imaginary unit.
Proof 1
\(\ds \csc i\) | \(=\) | \(\ds \frac 1 {\sin i}\) | Definition of Complex Cosecant Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\paren {\frac e 2 - \frac 1 {2 e} } i}\) | Sine of $i$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac 1 {\frac 1 {2 e} - \frac e 2} } i\) | Reciprocal of $i$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {2 e} {1 - e^2} } i\) | multiplying denominator and numerator by $2 e$ |
$\blacksquare$
Proof 2
\(\ds \csc i\) | \(=\) | \(\ds -i \csch 1\) | Hyperbolic Cosecant in terms of Cosecant | |||||||||||
\(\ds \) | \(=\) | \(\ds -\paren {\frac 2 {e^1 - e^{-1} } } i\) | Definition of Hyperbolic Cosecant | |||||||||||
\(\ds \) | \(=\) | \(\ds -\paren {\frac {2 e} {e^2 - 1} } i\) | multiplying denominator and numerator by $e$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {2 e} {1 - e^2} } i\) |
$\blacksquare$