Question

If $|\cot x+\text{cosec}x|=\left|\cot x\right|+\left|\text{cosec}x\right|;x\in [0,2\pi ]$, then complete set of values of $x$ is

• A. $[0,\pi ]$
• B. $(0,\frac{\pi }{2}]$
• C. $(0,\frac{\pi }{2}]\cup \left[\frac{3\pi }{2},2\pi \right)$
• D. $(\pi ,\frac{3\pi }{2}]\cup \left[\frac{7\pi }{4},2\pi \right)$

Answer 1

The correct option is C $(0,\frac{\pi }{2}]\cup \left[\frac{3\pi }{2},2\pi \right)$

This is possible only when
(1) $\cot x$ and $\csc x$ is positive i.e. $x\in (0,\frac{\pi }{2}]$
(2) $\cot x$ and $\csc x$ is negative i.e. $x\in \left[\frac{3\pi }{2},2\pi \right)$