# Derivative of Cotangent Function

## Theorem

$\displaystyle D_x \left({\cot x}\right) = -\csc^2 x = \frac {-1} {\sin^2 x}$, when $\sin x \ne 0$.

## Proof

Then:

 $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle D_x \left({\cot x}\right)$$ $$=$$ $$\displaystyle$$ $$\displaystyle \frac {\sin x \left({-\sin x}\right) - \cos x \cos x} {\sin^2 x}$$ $$\displaystyle$$ $$\displaystyle$$ Quotient Rule for Derivatives $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \frac {-\left({\sin^2 x + \cos^2 x}\right)} {\sin^2 x}$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \frac {-1} {\sin^2 x}$$ $$\displaystyle$$ $$\displaystyle$$ Sum of Squares of Sine and Cosine

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

The result follows from the definition of the cosecant function.

$\blacksquare$