Cosecant Function is Odd

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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

\(\displaystyle \map \csc {-x}\) \(=\) \(\displaystyle \frac 1 {\map \sin {-x} }\) 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