Answer
1) Rewrite $\cot x$ into $\frac{\cos x}{\sin x}$
2) Apply the Quotient Rule for $\frac{d}{dx}\Bigg(\frac{\cos x}{\sin x}\Bigg)$
3) Use trigonometrical rules to simplify.
Work Step by Step
We know that $\cot x=\frac{\cos x}{\sin x}$. So, $$\frac{d}{dx}(\cot x)=\frac{d}{dx}\Bigg(\frac{\cos x}{\sin x}\Bigg)$$
Now apply the Quotient Rule, we have $$\frac{d}{dx}(\cot x)=\frac{(\cos x)'\sin x-\cos x(\sin x)'}{(\sin x)^2}$$
$$\frac{d}{dx}(\cot x)=\frac{(-\sin x)\sin x-\cos x(\cos x)}{\sin^2 x}$$
$$\frac{d}{dx}(\cot x)=\frac{-\sin^2 x-\cos^2 x}{\sin^2 x}$$
$$\frac{d}{dx}(\cot x)=\frac{-(\sin^2 x+\cos^2 x)}{\sin^2 x}$$
$$\frac{d}{dx}(\cot x)=\frac{-1}{\sin^2 x}$$
$$\frac{d}{dx}(\cot x)=-\csc^2 x$$
The statement has been proved.
