Question

The number of solutions of the equation $|\cot x|=\cot x+\frac{1}{\sin x}(0\le x\le 2\pi )$ is :

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is A $0$

$\left(\cot x\right)=\cot x+\frac{1}{\sin x}$ $0\le x\le 2\pi$

$0\pi \le x<\frac{3\pi }{2}$ and $0\le x\le \frac{\pi }{2}$

$\cot x=\cot x+\frac{1}{\sin x}$

$\sin x=0$

$x=0$

$2\cot x=\frac{1}{\sin x}$

$\frac{2}{\sin x}\cos x=\frac{1}{\sin x}$

$\sin x=1\Rightarrow x=\frac{\pi }{2}$

$0\cos x=\frac{1}{2}$

$x=\frac{\pi }{3}$