If $\int_0^{\frac{\pi}{2}} \frac{\cot x}{\cot x+\operatorname{cosec} x} d x=m(\pi+n)$, then (m.n) equals

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Aicharya · Accepted Answer

$\begin{aligned} & \int_0^{\frac{\pi}{2}} \frac{\cot x}{\cot x+\operatorname{cosec} x} d x=\int_0^{\frac{\pi}{2}} \frac{\cos x}{\cos x+1} d x \ & =\int_0^{\frac{\pi}{2}} \frac{2 \cos ^2 \frac{x}{2}-1}{2 \cos ^2 \frac{x}{2}} d x=\int_0^{\frac{\pi}{2}}\left(1-\frac{1}{2} \sec ^2 \frac{x}{2}ight) d x \ & =\left[x-	an \frac{x}{2}ight]_0^{\pi / 2}=\frac{\pi}{2}-1=\frac{1}{2}(\pi+(-2)) \ & \Rightarrow m=\frac{1}{2} 	ext { and } n=-2 \ & \Rightarrow \mathrm{m} 	imes \mathrm{n}=\frac{1}{2} 	imes(-2)=-1\end{aligned}$

If $\int_0^{\frac{\pi}{2}} \frac{\cot x}{\cot x+\operatorname{cosec} x} d x=m(\pi+n)$, then (m.n) equals

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