# Derivative of Cosecant Function

## Theorem

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

where $\sin x \ne 0$.

## Proof 1

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 $\sin x \ne 0$.

$\blacksquare$

## Proof 2

 $\ds \map {\dfrac \d {\d x} } {\csc x}$ $=$ $\ds \map {\dfrac \d {\d x} } {\dfrac 1 {\sin x} }$ Definition of Real Cosecant Function $\ds$ $=$ $\ds \dfrac {\sin x \map {\frac \d {\d x} } 1 - 1 \map {\frac \d {\d x} }{\sin x} } {\sin^2 x}$ Quotient Rule for Derivatives $\ds$ $=$ $\ds \dfrac {0 - \cos x} {\sin^2 x}$ Derivative of Sine Function, Derivative of Constant $\ds$ $=$ $\ds -\csc x \cot x$ Definition of Real Cosecant Function, Definition of Real Cotangent Function

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

$\blacksquare$