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

 

 

 

  • Option 1)

    -1

  • Option 2)

    1

  • Option 3)

    -\frac{1}{2}

  • Option 4)

    \frac{1}{2}

 

Answers (1)

\int_{0}^{\frac{\pi }{2}}\frac{\cot x}{\cot x+\csc x}dx=\int_{0}^{\frac{\pi }{2}}\frac{\frac{\cos x}{\sin x}}{\frac{\cos x}{\sin x}+\frac{1}{\sin x}}dx

                                          =\int_{0}^{\frac{\pi }{2}}\frac{\cos x}{\cos x+1}dx

                                         =\int_{0}^{\frac{\pi }{2}}\frac{2\cos ^{2}\frac{x}{2}-1}{2\cos ^{2}\frac{x}{2}-x+x}dx

                                       =\int_{0}^{\frac{\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}\left [ \pi -2 \right ]

So, m=\frac{1}{2}         and     n=-2

\therefore mn=\frac{1}{2}\times -2=-1


Option 1)

-1

Option 2)

1

Option 3)

-\frac{1}{2}

Option 4)

\frac{1}{2}

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