Cosecant in terms of Hyperbolic Cosecant

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $z \in \C$ be a complex number.

Then:

$i \csc = -\csch \paren {i z}$

where:

$\csc$ denotes the cosecant function
$\csch$ denotes the hyperbolic cosecant
$i$ is the imaginary unit: $i^2 = -1$.


Proof

\(\ds i \csc x\) \(=\) \(\ds \frac i {\sin z}\) Definition of Complex Cosecant Function
\(\ds \) \(=\) \(\ds \frac 1 {-i \sin z}\) $i^2 = -1$
\(\ds \) \(=\) \(\ds \frac 1 {-\sinh \paren {i z} }\) Sine in terms of Hyperbolic Sine
\(\ds \) \(=\) \(\ds -\csch \paren {i z}\) Definition 2 of Hyperbolic Cosecant

$\blacksquare$


Also see


Sources