Derivative of Cotangent Function/Proof

Proof
From the definition of the cotangent function:
 * $\cot x = \dfrac {\cos x} {\sin x}$

From Derivative of Sine Function:
 * $\map {\dfrac \d {\d x} } {\sin x} = \cos x$

From Derivative of Cosine Function:
 * $\map {\dfrac \d {\d x} } {\cos x}= -\sin x$

Then:

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

The result follows from the definition of the real cosecant function.