Derivative of the Real Cotangent Function

Theorem: Derivative of the Real Cotangent Function

The real cotangent function is differentiable (everywhere it is defined) and its derivative is the negative reciprocal of the square of the real sine function:

$$(\cot x{)}^{\prime }=-\frac{1}{{\sin }^{2}x}=-1-{\cot }^{2}x$$

PROOF

$$(\cot x{)}^{\prime }={\left(\frac{\cos x}{\sin x}\right)}^{\prime }=\frac{(\cos x{)}^{\prime }\sin x-(\sin x{)}^{\prime }\cos x}{{\sin }^{2}x}=\frac{-{\sin }^{2}x-{\cos }^{2}x}{{\sin }^{2}x}=\frac{-({\sin }^{2}x+{\cos }^{2}x)}{{\sin }^{2}x}=\frac{-1}{{\sin }^{2}x}$$