\(sinx+cosx\cdot sin2x+\sqrt{3}cos3x=2.\left(cos4x+sin^3x\right)\)
\(\Leftrightarrow sinx+cosx\cdot sin2x+\sqrt{3}cos3x=2cos4x+2sin^3x\)
\(\Leftrightarrow sinx-2sin^3x+cosx.sin2x+\sqrt{3}cos3x=2cos4x\)
\(\Leftrightarrow sinx.\left(1-2sin^2x\right)+cosx.sin2x+\sqrt{3}cos3x=2cos4x\)
\(\Leftrightarrow sinx.cos2x+cosx.sin2x+\sqrt{3}cos3x=2cos4x\)
\(\Leftrightarrow sin.\left(x+2x\right)+\sqrt{3}cos3x=2cos4x\)
\(\Leftrightarrow sin3x+\sqrt{3}cos3x=2cos4x\)
\(\Leftrightarrow\dfrac{1}{2}sin3x+\dfrac{\sqrt{3}}{2}cos3x=cos4x\)
\(\Leftrightarrow cos\dfrac{\pi}{3}.sin3x+sin\dfrac{\pi}{3}.cos3x=cos4x\)
\(\Leftrightarrow sin.\left(3x+\dfrac{\pi}{3}\right)=sin\left(\dfrac{\pi}{x}-4x\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}3x+\dfrac{\pi}{3}=\dfrac{\pi}{2}-4x+k2\pi\\3x+\dfrac{\pi}{2}=\pi-\dfrac{\pi}{2}+4x+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{42}+\dfrac{k2\pi}{7}\\x=-\dfrac{\pi}{6}+k2\pi\end{matrix}\right.\left(k\in Z\right)\)
