Mathematics - Definite Integration Question with Solution | TestHub

Let $I=\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\frac{\cot x}{\cot x+\mathrm{cosec}x}dx$
$\Rightarrow I=\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\frac{\mathrm{c}\mathrm{o}\mathrm{s}x}{\mathrm{c}\mathrm{o}\mathrm{s}x+1}dx$
$\Rightarrow I=\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\frac{\mathrm{c}\mathrm{o}\mathrm{s}x+1-1}{\mathrm{c}\mathrm{o}\mathrm{s}x+1}dx$
$\Rightarrow I=\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\left(1-\frac{1}{2{\cos }^2\frac{x}{2}}\right)dx$
$\Rightarrow I=\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\left(1-\frac{1}{2}\mathrm{s}\mathrm{e}{\mathrm{c}}^2\frac{x}{2}\right)dx$
$\Rightarrow I={\left(x-\tan \frac{x}{2}\right)}_0^{\frac{\mathrm{\pi }}{2}}$
$\Rightarrow I=\left(\frac{\mathrm{\pi }}{2}-1\right)-\left(0-0\right)$
$\Rightarrow I=\frac{\mathrm{\pi }}{2}-1=\frac{1}{2}\left(\mathrm{\pi }-2\right)$.

Now using given information $\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\frac{\cot x}{\cot x+\mathrm{cosec}x}dx=m(\mathrm{\pi }+n),$ clearly $m=\frac{1}{2}$ and $n=-2$.

$\Rightarrow mn=-1$