Question

Solve:
$\int \frac{dx}{{\sin }^{2}x{\cos }^{2}x}$

Answer 1

$\int \frac{dx}{{\sin }^{2}x{\cos }^{2}x}$

Multiply and divide the above expression by $4$, we haave

$=\int \frac{4dx}{4{\sin }^{2}x{\cos }^{2}x}$

$=4\int \frac{dx}{{\left(2\sin x\cos x\right)}^2}$

$=4\int \frac{dx}{{\sin }^{2}2x}[\because \sin 2x=2\sin x\cos x]$

$=4\int {\csc }^{2}2xdx$

$=4\left(\frac{-\cot 2x}{2}\right)+C[\because \int {\csc }^{2}axdx=\frac{-\cot ax}{a}]$

$=-2\cot 2x+C$

$\therefore \int \frac{dx}{{\sin }^{2}x{\cos }^{2}x}=-2\cot 2x+C$