Question

Evaluate: ∫ (cos 2x/ cos² x .sin² x) dx

Answer 1

\(\int \frac{\cos 2x}{\cos^2x\sin^2x}dx\)

\(= \int \frac{\cos^2x - \sin^2x}{cos^2x\sin^2x} dx\)

\(= \int \frac{\cos^2x}{cos^2x\sin^2x} dx - \int \frac{\sin^2x}{cos^2x\sin^2x}dx\)

\(= \int \frac 1{\sin^2 x}dx - \int\frac 1{\cos^2 x}dx\)

\(= \int cosec^2x \,dx - \int \sec^2x\, dx\)

\(= -\cot x - \tan x + c\)

Answer 2

= ∫ (cosec² x – sec² x) dx 

= – (cot x + tan x) + c