A) \[-1\]
B) \[1\]
C) \[\frac{1}{2}\]
D) \[-\frac{1}{2}\]
Correct Answer: A
Solution :
\[\int\limits_{0}^{\pi /2}{\frac{\cot \,xdx}{\cot \,x+cosecx}}\] \[\int\limits_{0}^{\pi /2}{\frac{\cot \,x}{cos\,x+1}=\int_{{}}^{{}}{\frac{2{{\cos }^{2}}\frac{x}{2}-1}{2{{\cos }^{2}}\frac{x}{2}}}}\] \[\int\limits_{0}^{\pi /2}{\left( 1-\frac{1}{2}{{\sec }^{2}}\frac{\,x}{2} \right)dx}\]\[\left[ x-\tan \frac{x}{2} \right]_{0}^{\frac{\pi }{2}}\] \[\frac{1}{2}[\pi -2]\] \[m=\frac{1}{2},n=-2\] \[mn=-1\]
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