\(\dfrac{1}{sin2x}+\dfrac{1}{cos2x}=\dfrac{2}{sin4x}\)
\(ĐK:\left\{{}\begin{matrix}sin2x\ne0\\cos2x\ne0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\ne\dfrac{k\pi}{2}\\x\ne\dfrac{\pi}{4}+\dfrac{k\pi}{2}\end{matrix}\right.\) \(\left(k\in Z\right)\)
\(\Leftrightarrow\dfrac{sin2x+cos2x}{sin2x.cos2x}=\dfrac{2}{2\left(sin2x.cos2x\right)}\)
\(\Leftrightarrow\dfrac{sin2x+cos2x}{sin2x.cos2x}=\dfrac{1}{sin2x.cos2x}\)
\(\Leftrightarrow sin2x+cos2x=1\)
\(\Leftrightarrow\dfrac{1}{\sqrt{2}}sin2x+\dfrac{1}{\sqrt{2}}cos2x=\dfrac{1}{\sqrt{2}}\)
\(\Leftrightarrow sin2x.cos\dfrac{1}{4}\pi+cos2x.sin\dfrac{1}{4}\pi=\dfrac{1}{\sqrt{2}}\)
\(\Leftrightarrow sin\left(2x+\dfrac{1}{4}\pi\right)=sin\dfrac{1}{4}\pi\)
\(\Leftrightarrow\left[{}\begin{matrix}2x+\dfrac{\pi}{4}=\dfrac{\pi}{4}+k2\pi\\2x+\dfrac{\pi}{4}=\dfrac{3}{4}\pi+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=k\pi\\x=\dfrac{1}{4}\pi+k\pi\end{matrix}\right.\) `(tm)`
