Giải phương trình \(\sin x + \cos x = 2\sqrt 2 \sin x\cos x\).

\(\sin x + \cos x = 2\sqrt 2 \sin x\cos x\)

\( \Leftrightarrow \sqrt 2 \sin \left( {x + \frac{\pi }{4}} \right) = \sqrt 2 \sin 2x \Leftrightarrow \sin \left( {x + \frac{\pi }{4}} \right) = \sin 2x\)

\( \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{2x = x + \frac{\pi }{4} + k2\pi }\\{2x = \pi - x - \frac{\pi }{4} + k2\pi }\end{array}} \right. \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{x = \frac{\pi }{4} + k2\pi }\\{x = \frac{\pi }{4} + k\frac{{2\pi }}{3}}\end{array}} \right.\left( {k \in \mathbb{Z}} \right)\)