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:

$\csc \left({-x}\right) = -\csc x$

That is, the cosecant function is odd.


Proof

\(\displaystyle \csc \left({-x}\right)\) \(=\) \(\displaystyle \frac 1 {\sin \left({-x}\right)}\) Cosecant is Reciprocal of Sine
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {- \sin x}\) Sine Function is Odd
\(\displaystyle \) \(=\) \(\displaystyle - \csc x\) Cosecant is Reciprocal of Sine

$\blacksquare$


Also see


Sources