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\(\Leftrightarrow\frac{2sin4x.cos2x}{cos2x}-2cos4x=2\sqrt{2}\)
\(\Leftrightarrow2sin4x-2cos4x=2\sqrt{2}\)
\(\Leftrightarrow sin4x-cos4x=\sqrt{2}\)
\(\Leftrightarrow\sqrt{2}sin\left(4x-\frac{\pi}{4}\right)=\sqrt{2}\)
\(\Leftrightarrow sin\left(4x-\frac{\pi}{4}\right)=1\)
\(\Leftrightarrow4x-\frac{\pi}{4}=\frac{\pi}{2}+k2\pi\)
\(\Rightarrow x=\frac{3\pi}{16}+\frac{k\pi}{2}\)
d/
Đặt \(sin2x-cos2x=\sqrt{2}sin\left(2x-\frac{\pi}{4}\right)=t\Rightarrow\left|t\right|\le\sqrt{2}\)
\(\Rightarrow t^2-3t-4=0\Rightarrow\left[{}\begin{matrix}t=-1\\t=4\left(l\right)\end{matrix}\right.\)
\(\Rightarrow\sqrt{2}sin\left(2x-\frac{\pi}{4}\right)=-1\)
\(\Leftrightarrow sin\left(2x-\frac{\pi}{4}\right)=-\frac{\sqrt{2}}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}2x-\frac{\pi}{4}=-\frac{\pi}{4}+k2\pi\\2x-\frac{\pi}{4}=\frac{5\pi}{4}+k2\pi\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=k\pi\\x=\frac{3\pi}{4}+k\pi\end{matrix}\right.\)
1.
\(\Leftrightarrow cos3x+sin3x-2sin3x.cos3x=0\)
\(\Leftrightarrow cos3x+sin3x-\left(2sin3x.cos3x+1\right)+1=0\)
\(\Leftrightarrow cos3x+sin3x-\left(sin3x+cos3x\right)^2+1=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin3x+cos3x=\frac{\sqrt{5}+1}{2}\\sin3x+cos3x=\frac{1-\sqrt{5}}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}sin\left(3x+\frac{\pi}{4}\right)=\frac{\sqrt{10}+\sqrt{2}}{4}>1\left(l\right)\\sin\left(3x+\frac{\pi}{4}\right)=\frac{\sqrt{2}-\sqrt{10}}{4}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}3x+\frac{\pi}{4}=arcsin\left(\frac{\sqrt{2}-\sqrt{10}}{4}\right)+k2\pi\\3x+\frac{\pi}{4}=\pi-arcsin\left(\frac{\sqrt{2}-\sqrt{10}}{4}\right)+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow x=...\)
2.
\(\Leftrightarrow sinx-\left(1+cosx\right)+sin2x=-2\)
\(\Leftrightarrow sinx-cosx+1+sin2x=0\)
\(\Leftrightarrow sinx-cosx-\left(1-2sinx.cosx\right)+2=0\)
\(\Leftrightarrow sinx-cosx-\left(sinx-cosx\right)^2+2=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx-cosx=-1\\sinx-cosx=2\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}sin\left(x-\frac{\pi}{4}\right)=-\frac{\sqrt{2}}{2}\\sin\left(x-\frac{\pi}{4}\right)=\sqrt{2}>1\left(l\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x-\frac{\pi}{4}=-\frac{\pi}{4}+k2\pi\\x-\frac{\pi}{4}=\frac{5\pi}{4}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow x=...\)
\(\cos3x+2\sin2x-\cos x=0\Leftrightarrow2\sin2x\left(1-\sin x\right)=0\)
\(\Leftrightarrow\begin{cases}\sin2x=0\\\sin x=0\end{cases}\)
\(\Leftrightarrow\begin{cases}x=k\frac{\pi}{2}\\x=\frac{\pi}{2}+k2\pi\end{cases}\)