Question

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

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (C)

(i) When $x \in\left[0, \frac{\pi}{2}\right] \cup \left( \pi , \frac{3 \pi}{2} \right)$ then $\cot \, x \geq 0$
$ \Rightarrow \:\: \cot \, x = \cot \, x + \frac{1}{\sin \, x} \:\:\: \Rightarrow \frac{1}{\sin \, x} = 0$
$ \Rightarrow $ No solution exist
(ii) When $x \in\left(\frac{\pi}{2}, \pi \right) \cup \left( \frac{3 \pi}{2} , 2\pi \right)$ then $\cot \, x < 0$
$ \therefore \:\: - \cot \, x = \cot \, x + \frac{1}{\sin \, x}$
$ \Rightarrow \:\: -2 \, \cot \, x = \frac{1}{\sin \, x}$
$ \Rightarrow \:\: \cos \, x = \frac{-1}{2 } \:\: \Rightarrow \: x = \frac{2 \pi}{3}$