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\[\begin{array}{l} 2{\cos ^2}x = \sqrt 3 \\ \Leftrightarrow 1 + \cos 2x = \sqrt 3 \\ \Leftrightarrow \cos 2x = \sqrt 3 - 1\\ \Leftrightarrow \left[ \begin{array}{l} 2x = \arccos \left( {\sqrt 3 - 1} \right) + k2\pi \\ 2x = - \arccos \left( {\sqrt 3 - 1} \right) + k2\pi \end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l} x = \frac{1}{2}\arccos \left( {\sqrt 3 - 1} \right) + k\pi \\ x = - \frac{1}{2}\arccos \left( {\sqrt 3 - 1} \right) + k\pi \end{array} \right.\,\,\,\,\left( {k \in Z} \right). \end{array}\]
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