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\(\begin{array}{l} {\sin ^2}\left( {5x + \frac{{2\pi }}{5}} \right) - {\cos ^2}\left( {\dfrac{x}{4} + \pi } \right) = 0\\ \Leftrightarrow 1 - \cos \left( {10x + \dfrac{{4\pi }}{5}} \right) - \left( {1 + \cos \left( {\dfrac{x}{2} + 2\pi } \right)} \right) = 0\\ \Leftrightarrow \cos \left( {10x + \dfrac{{4\pi }}{5}} \right) = \cos \dfrac{x}{2}\\ \Leftrightarrow \left[ \begin{array}{l} 10x + \dfrac{{4\pi }}{5} = \dfrac{x}{2} + k2\pi \\ 10x + \dfrac{{4\pi }}{5} = - \dfrac{x}{2} + k2\pi \end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l} x = \dfrac{{ - 8\pi }}{{95}} + \dfrac{{k4\pi }}{{19}}\\ x = \dfrac{{ - 8\pi }}{{105}} + \dfrac{{k4\pi }}{{21}} \end{array} \right. \end{array}\)
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