Question

Total number of solutions of $\mid \cot x\mid =\cot x+\frac{1}{\sin x},xϵ[0,3\pi ]$ is equal to-

• A. $1$
• B. $2$
• C. $3$
• D. Zero

Answer 1

Answer: (B)

$2$

Take cases $\cot x\ge 0\therefore \frac{1}{\sin x}=0$ (not possible)
$cotx<0\therefore -2\cot x=\frac{1}{\sin x}$
Now the only possible solution is
$\cos x=\frac{-1}{2}$
$x=\frac{2\pi }{3},\frac{4\pi }{3},\frac{8\pi }{3}$
But $x=\frac{4\pi }{3}$ lies in 3rd quadrant in which $|\cot x|=+\cot x$
So only two possible solutions are there .