Answer
$$\int\frac{\cot x}{\cos^2x}dx=-\ln|\csc2x+\cot2x|+C$$
Work Step by Step
$$A=\int\frac{\cot x}{\cos^2x}dx=\int\frac{\frac{\cos x}{\sin x}}{\cos^2x}dx$$ $$A=\int\frac{\cos x}{\sin x\cos^2x}dx=\int\frac{1}{\sin x\cos x}dx$$ $$A=\int\frac{1}{\frac{1}{2}\times2\sin x\cos x}dx$$ $$A=2\int\frac{1}{\sin2x}dx$$ $$A=2\int\csc2xdx$$ $$A=2\times\frac{1}{2}(-\ln|\csc2x+\cot2x|)+C$$ $$A=-\ln|\csc2x+\cot2x|+C$$
