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Đặt $t= \tan x+ \cot x \Rightarrow t^2=\tan^2x +\cot^2 x+2\Rightarrow \tan^2x +\cot^2 x=t^2-2.$
PT
$\Leftrightarrow 3(t^2-2)+4t+2=0\Leftrightarrow 3t^2+4t-4=0\Leftrightarrow t=-2$ hoặc $t=2/3.$
+ Nếu $t=-2\Leftrightarrow \tan x+ \dfrac{1}{\tan x}+2=0\Leftrightarrow (\tan x+1)^2=0\Leftrightarrow \tan x=-1.$
+ Nếu $t=\dfrac{2}{3}\Leftrightarrow \tan x+ \dfrac{1}{\tan x}-\dfrac{2}{3}=0\Leftrightarrow3\tan^2x-2\tan x+3=0$, Pt này vô nghiệm.
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