<s>[INST] Evaluate the integrals $\int \sin{x} \cot^2{x} \,dx$ and $\int \cos{x} \cot^2{x} \,dx$. Can you please show how to evaluate the integrals $$\int \sin{x} \cot^2{x} \,dx$$ and $$\int \cos{x} \cot^2{x} \,dx.$$
I know that $\cot x=\dfrac{\cos x}{\sin x}$, which can simplify the integrands a bit. 
$$\sin x\cot^2x = \frac{\cos^2x}{\sin x}\\
\cos x\cot^2x = \frac{\cos^3x}{\sin^2 x}$$
But I still do not know where to continue from there.
 [/INST] HINT:
$$\sin x\cot^2x=\frac{\cos^2x}{\sin x}=\frac{1-\sin^2x}{\sin x}=\csc x-\sin x$$
$$\cos x\cot^2x=\frac{\cos^3x}{\sin^2x}=\frac{\cos^3x-\cos x+\cos x}{1-\cos^2x}=-\cos x+\frac{\cos x}{\sin^2x}$$
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