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 $e$
\(\ds \) \(=\) \(\ds \paren {\frac {2 e} {1 - e^2} } i\)

$\blacksquare$


Also see