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