\[\begin{array}{l}
si{n^2}(x - \frac{\pi }{4}) = co{s^2}x\\
\Leftrightarrow {\sin ^2}\left( {x - \frac{\pi }{4}} \right) = {\left( {\sin \left( {\frac{\pi }{2} - x} \right)} \right)^2}\\
\Leftrightarrow \left[ \begin{array}{l}
\sin \left( {x - \frac{\pi }{4}} \right) = \sin \left( {\frac{\pi }{2} - x} \right)\\
\sin \left( {x - \frac{\pi }{4}} \right) = - \sin \left( {\frac{\pi }{2} - x} \right)
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x - \frac{\pi }{4} = \frac{\pi }{2} - x + k2\pi \\
x - \frac{\pi }{4} = \pi - \frac{\pi }{2} + x + k2\pi \\
\sin \left( {x - \frac{\pi }{4}} \right) = \sin \left( {x - \frac{\pi }{2}} \right)
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = \frac{{3\pi }}{8} + k\pi \\
x - \frac{\pi }{4} = x - \frac{\pi }{2} + k2\pi \\
x - \frac{\pi }{4} = \frac{\pi }{2} - x + k2\pi
\end{array} \right.\\
\Leftrightarrow x = \frac{{3\pi }}{8} + k\pi \left( {k \in Z} \right)\\
vay:S = \{ \frac{{3\pi }}{8} + k\pi \}
\end{array}\]