Question

Use the definition of the cotangent function and the quotient rule to prove if f(x)=cotx, than f'(x)= -cosec²x.

Answer 1

$$f'(x)=(\cot x)'=(\dfrac{\cos x}{\sin x})'$$

$$=\dfrac{(\cos x)'\sin x-\cos x(\sin x)'}{\sin^2x}=\dfrac{-\sin^2x-\cos^2x}{\sin^2x}$$

$$=\dfrac{-(\sin^2x+\cos^2x)}{\sin^2x}=\dfrac{-1}{\sin^2x}=-\cosec^2x$$