Question

The number of solution of the equation $|\cot x|=cotx+\frac{1}{sinx}$ in $[0,2\pi ]$ is

• A. $2$
• B. $4$
• C. $0$
• D. $1$

Answer 1

The correct option is D $1$

Let split the domain in two parts $[0,\frac{\pi }{2}]\cup [\pi ,\frac{3\pi }{2}]and[\frac{\pi }{2},\pi ]\cup [\frac{3\pi }{2},2\pi ]$
Case -1
$xϵ[0,\frac{\pi }{2}]\cup [\pi ,\frac{3\pi }{2}]$
$|cotx|=cotx=cotx+\frac{1}{sinx}$

$\frac{1}{sinx}=0$
$\Rightarrow$ no solution
Case -2
$xϵ[\frac{\pi }{2},\pi ]\cup [\frac{3\pi }{2},2\pi ]$
$|cotx|=-cotx=cotx+\frac{1}{sinx}$
$\frac{2cosx+1}{sinx}=0$
$cosx=-\frac{1}{2}$
only 1 solution i.e. $x={120}^o$