Question

If $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\cot x}{\cot x+\text{cosec}x}dx=m(\pi +n),$ then $m.n$ is equal to :

• A. $1$
• B. $-1$
• C. $\frac{1}{2}$
• D. $-\frac{1}{2}$

Answer 1

The correct option is B $-1$

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\cot x}{\cot x+\text{cosec}x}dx=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\cos x}{\cos x+1}dx$
$=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{2{\cos }^2\frac{x}{2}-1}{2{\cos }^2\frac{x}{2}}dx=\underset{0}{\overset{\frac{\pi }{2}}{\int }}(1-\frac{1}{2}{\sec }^2\frac{x}{2})dx$
$=\frac{\pi }{2}-\frac{2}{2}{\tan \frac{x}{2}|}_0^{\frac{\pi }{2}}$
$=\frac{\pi }{2}-\tan \frac{\pi }{4}=\frac{1}{2}\cdot (\pi -2)=m(\pi +n)$
$\therefore m\cdot n=\frac{1}{2}\times -2=-1$