Question

The number of solution of the equation $3{\tan }^{2}x-4\sqrt{3}|\tan x|+4\sec x+11=0$ for $x\in [0,2\pi ]$ is

Answer 1

$3{\tan }^{2}x-4\sqrt{3}|\tan x|+4\sec x+11=0$
$\Rightarrow 2{\tan }^{2}x+{\tan }^{2}x+1-4\sqrt{3}|\tan x|+4\sec x+10=0$
$\Rightarrow 2({\tan }^{2}x-2\sqrt{3}|\tan x|+3)+({\sec }^{2}x+4\sec x+4)=0$
$\Rightarrow 2(|\tan x|-\sqrt{3}{)}^2+(\sec x+2{)}^2=0$
This is possible only when
$\Rightarrow |\tan x|=\sqrt{3}\text{and}\sec x+2=0$
$\Rightarrow \tan x=\pm \sqrt{3}\text{and}\sec x=-2$
$\Rightarrow \tan x=\pm \sqrt{3}\text{and}\cos x=-\frac{1}{2}\Rightarrow x=\frac{\pi }{3},\frac{2\pi }{3},\frac{4\pi }{3},\frac{5\pi }{3}\text{and}x=\frac{2\pi }{3},\frac{4\pi }{3}$
$\therefore x=\frac{2\pi }{3},\frac{4\pi }{3}$

Hence, the number of solutions is $2$.