Question

Total number of solutions of $\tan x+\cot x=2\text{cosec}x$ , where $x\in [-2\pi ,2\pi ]$ is :

• A. $5$
• B. $4$
• C. $6$
• D. $3$

Answer 1

The correct option is B $4$

$\tan x+\cot x=2\text{cosec}x$
$\Rightarrow \frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}=\frac{2}{\sin x}\Rightarrow \frac{1}{\sin x\cos x}=\frac{2}{\sin x}\Rightarrow \cos x=\frac{1}{2}[\sin x\ne 0]\Rightarrow x=\pm \frac{\pi }{3},\pm \frac{5\pi }{3}$
$\therefore$ There are four solutions.