Prove the following


If $\int_{0}^{\frac{\pi}{2}} \frac{\cot x}{\cot x+\operatorname{cosec} x} d x=m(\pi+n)$, then $m \cdot n$ is equal to :

  1. $-1$

  2. 1

  3. $\frac{1}{2}$

  4. $-\frac{1}{2}$

Correct Option: 1


$\int_{0}^{\pi / 2} \frac{\cot x d x}{\cot x+\cos e c x}$

$\int_{0}^{\pi / 2} \frac{\cos x}{\cos x+1}=\int \frac{2 \cos ^{2} \frac{x}{2}-1}{2 \cos ^{2} \frac{x}{2}}$

$\int_{0}^{\pi / 2}\left(1-\frac{1}{2} \sec ^{2} \frac{x}{2}\right) d x$

$\left[x-\tan \frac{x}{2}\right]_{0}^{\frac{\pi}{2}}$

$\frac{1}{2}[\pi-2]$                    $\mathrm{m}=\frac{1}{2}, \quad \mathrm{n}=-2$

$m n=-1$

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