`sin2x + cos4x =0`

`<=>sin2x + 1 - 2sin^2 2x=0`

`<=>2sin^2 2x - sin2x -1=0`

`<=>2sin^2 2x + sin2x -2sin2x-1=0`

`<=>sin2x(2sin2x+1)-(2sin2x+1)=0`

`<=>(sin2x -1)(2sin2x+1)=0`

`<=>`\(\left[ \begin{array}{l}sin2x=1\\sin2x= \dfrac{-1}{2}\end{array} \right.\) 

`<=>`\(\left[ \begin{array}{l}2x=\dfrac{\pi}{2}+k2\pi,k \in Z\\2x= \dfrac{-\pi}{6}+h2pi, h \in Z\\2x= \pi -\dfrac{-\pi}{6}+h2pi,h \in Z\end{array} \right.\) 

`<=>`\(\left[ \begin{array}{l}x=\dfrac{\pi}{4}+k\pi,k \in Z\\x= \dfrac{-\pi}{12}+hpi, h \in Z\\x= \dfrac{7\pi}{12}+hpi,h \in Z\end{array} \right.\)