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Điều kiện:$x \ne k\pi $
$\begin{array}{l}
(*) \Leftrightarrow \frac{{3{{\cos }^2}x}}{{{{\sin }^2}x}} + 2\sqrt 2 {\sin ^2}x = (2 + 3\sqrt 2 )\cos x\\
\Leftrightarrow 3{\cos ^2}x - 3\sqrt 2 {\sin ^2}x.\cos x + 2\sqrt 2 {\sin ^4}x - 2{\sin ^2}x\cos x = 0\\
\Leftrightarrow 3\cos x\left( {\cos x - \sqrt 2 {{\sin }^2}x} \right) - 2{\sin ^2}x\left( {\cos x - \sqrt 2 {{\sin }^2}x} \right) = 0\\
\Leftrightarrow \left( {\cos x - \sqrt 2 {{\sin }^2}x} \right)\left( {3\cos x - 2{{\sin }^2}x} \right) = 0\\
\Leftrightarrow (\cos x - \sqrt 2 + \sqrt 2 c{\rm{o}}{{\rm{s}}^2}x)(3\cos x - 2 + 2{\cos ^2}x) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\sqrt 2 c{\rm{o}}{{\rm{s}}^2}x + \cos x - \sqrt 2 = 0\\
2{\cos ^2}x + 3\cos x - 2 = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\cos x = \frac{{ - 1 + \sqrt 3 }}{{\sqrt 2 }}\\
\cos x = \frac{1}{2}
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = \pm \alpha + k2\pi \\
x = \pm \frac{\pi }{3} + k2\pi
\end{array} \right.\,\,(c{\rm{os}}\alpha = \frac{{ - 1 + \sqrt 3 }}{{\sqrt 2 }},\,\,k \in Z)
\end{array}$
So sánh với điều kiện ta có nghiệm cần tìm là: $\left[ \begin{array}{l}
x = \pm \alpha + k2\pi \\
x = \pm \frac{\pi }{3} + k2\pi
\end{array} \right.\,\,(c{\rm{os}}\alpha = \frac{{ - 1 + \sqrt 3 }}{{\sqrt 2 }},\,\,k \in Z)$
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