Cosecant Function is Odd

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $x \in \R$ be a real number.

Let $\csc x$ be the cosecant of $x$.


Then, whenever $\csc x$ is defined:

$\map \csc {-x} = -\csc x$

That is, the cosecant function is odd.


Proof

\(\ds \map \csc {-x}\) \(=\) \(\ds \frac 1 {\map \sin {-x} }\) Cosecant is Reciprocal of Sine
\(\ds \) \(=\) \(\ds \frac 1 {-\sin x}\) Sine Function is Odd
\(\ds \) \(=\) \(\ds -\csc x\) Cosecant is Reciprocal of Sine

$\blacksquare$


Also see


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Cosecant_Function_is_Odd&oldid=457325"