# Derivative of Cosecant Function

## Theorem

$\map {\dfrac \d {\d x} } {\csc x} = -\csc x \cot x$

where $\sin x \ne 0$.

## Proof

From the definition of the cosecant function:

$\csc x = \dfrac 1 {\sin x}$
$\map {\dfrac \d {\d x} } {\sin x} = \cos x$

Then:

 $\ds \map {\dfrac \d {\d x} } {\csc x}$ $=$ $\ds \cos x \frac {-1} {\sin^2 x}$ Chain Rule for Derivatives $\ds$ $=$ $\ds \frac {-1} {\sin x} \frac {\cos x} {\sin x}$ $\ds$ $=$ $\ds -\csc x \cot x$ Definition of Real Cosecant Function and Definition of Real Cotangent Function

This is valid only when $\sin x \ne 0$.

$\blacksquare$