The number of solutions of the equation

Question:

The number of solutions of the equation

$|\cot x|=\cot x+\frac{1}{\sin x}$ in the interval $[0,2 \pi]$ is

 

Solution:

If $\cot x>0 \Rightarrow \frac{1}{\sin x}=0$ (Not possible)

If $\cot x<0 \Rightarrow 2 \cot x+\frac{1}{\sin x}=0$

$\Rightarrow 2 \cos x=-1$

$\Rightarrow x=\frac{2 \pi}{3}$ or $\frac{4 \pi}{3}$ (reject)

 

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