Cotangent of i
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Theorem
- $\cot i = \paren {\dfrac {1 + e^2} {1 - e^2} } i$
where $\cot$ denotes the complex cotangent function and $i$ is the imaginary unit.
Proof 1
| \(\ds \cot i\) | \(=\) | \(\ds \frac {\cos i} {\sin i}\) | Definition of Complex Cotangent Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\frac e 2 + \frac 1 {2 e} } {\left({\frac e 2 - \frac 1 {2 e} }\right) i}\) | Cosine of $i$ and Sine of $i$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \left({\frac {e + \frac 1 e} {e - \frac 1 e} }\right) \left({\frac 1 i}\right)\) | multiplying denominator and numerator by $2$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \left({ \frac {e^2 + 1} {e^2 - 1} }\right) \left({\frac 1 i}\right)\) | multiplying denominator and numerator by $e$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \left({ \frac {1 + e^2} {1 - e^2} }\right) i\) | Reciprocal of $i$ |
$\blacksquare$
Proof 2
| \(\ds \cot i\) | \(=\) | \(\ds -i \coth 1\) | Hyperbolic Cotangent in terms of Cotangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\paren {\frac {e^1 + e^{-1} } {e^1 - e^{-1} } } i\) | Definition 1 of Hyperbolic Cotangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\paren {\frac {e^2 + 1} {e^2 - 1} } i\) | multiplying denominator and numerator by $e$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\frac {1 + e^2} {1 - e^2} } i\) |
$\blacksquare$