If int_0^{frac{pi}{2}} frac{cot x}{cot x + cs | vedclass

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(D) Let $I = \int_0^{\frac{\pi}{2}} \frac{\cot x}{\cot x + \csc x} dx$.Simplify the integrand: $\frac{\cot x}{\cot x + \csc x} = \frac{\frac{\cos x}{\

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$-1$

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(D) Let $I = \int_0^{\frac{\pi}{2}} \frac{\cot x}{\cot x + \csc x} dx$.Simplify the integrand: $\frac{\cot x}{\cot x + \csc x} = \frac{\frac{\cos x}{\sin x}}{\frac{\cos x}{\sin x} + \frac{1}{\sin x}} = \frac{\cos x}{\cos x + 1}$.Using the identity $\cos x = 2 \cos^2 \frac{x}{2} - 1$,we get $\frac{\cos x}{\cos x + 1} = \frac{2 \cos^2 \frac{x}{2} - 1}{2 \cos^2 \frac{x}{2}} = 1 - \frac{1}{2} \sec^2 \frac{x}{2}$.Now integrate: $I = \int_0^{\frac{\pi}{2}} (1 - \frac{1}{2} \sec^2 \frac{x}{2}) dx$.$I = [x - 	an \frac{x}{2}]_0^{\frac{\pi}{2}}$.Evaluating at the limits: $I = (\frac{\pi}{2} - 	an \frac{\pi}{4}) - (0 - 	an 0) = \frac{\pi}{2} - 1$.We can write this as $I = \frac{1}{2}(\pi - 2) = \frac{1}{2}(\pi + (-2))$.Comparing with $m(\pi + n)$,we get $m = \frac{1}{2}$ and $n = -2$.Therefore,$m \cdot n = \frac{1}{2} 	imes (-2) = -1$.

If $\int_0^{\frac{\pi}{2}} \frac{\cot x}{\cot x + \csc x} dx = m(\pi + n)$,then $m \cdot n$ is equal to

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