Question

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

Answer 1

Let $sin2x=t$

$2sinx.cosx=t$

$si{n}^{2}xco{s}^{2}x=\frac{t^2}{4}$

$\frac{d}{dx}\left(sin2x\right)=2.cos2x$

$\frac{dt}{dx}=2(co{s}^{2}x-si{n}^{2}x)$

$\Rightarrow \frac{dt}{2}=(co{s}^{2}x-si{n}^{2}x)dx$

the answer to given question is as, $\int \frac{co{s}^{2}x-si{n}^{2}x}{si{n}^{2}xco{s}^{2}x}dx=\int \frac{4}{t^2}.\frac{1}{2}.dt$

$\int 2.t^{-2}dt=\frac{-2}{t}+c=\frac{-2}{sin2x}+c$