Question

The complete solution set of the equation $1+2\text{cosec}x=-\frac{{\sec }^2\frac{x}{2}}{2}$ is

• A. $2n\pi -\frac{\pi }{2},nϵI$
• B. $n\pi -\frac{\pi }{2},nϵI$
• C. $2n\pi +\frac{\pi }{2},nϵI$
• D. $n\pi +\frac{\pi }{2},nϵI$

Answer 1

The correct option is A $2n\pi -\frac{\pi }{2},nϵI$

With $t=\tan \left(\frac{x}{2}\right),$

$\sin x=\frac{2t}{1+t^2}$ and ${\sec }^2\frac{x}{2}=1+t^2$

Now the given equation changes to

$1+2\left(\frac{1+t^2}{2t}\right)+\frac{1}{2}(1+t^2)=0$giving,

$t^3+2t^2+3t+2=(t+1)(t^2+t+2)=0$

solving,

$t=-1\Rightarrow \tan \frac{x}{2}=-1=\tan (-\frac{\pi }{4})$

that gives the solution in multitude as

$\frac{x}{2}=k\pi -\frac{\pi }{4}\Rightarrow x=2\pi k-\frac{\pi }{2}$

$k=0,\pm 1,\pm 2,.....$

Replacing k by n,

$x=2\pi n-\frac{\pi }{2}$

$n=0,\pm 1,\pm 2,.....$