Question

\(\sin4x=\sqrt{3}\left(1-\cos4x\right)\)

Answer 1

\(\Leftrightarrow sin4x+\sqrt{3}cos4x=\sqrt{3}\)

\(\Leftrightarrow\dfrac{1}{2}sin4x+\dfrac{\sqrt{3}}{2}cos4x=\dfrac{\sqrt{3}}{2}\)

\(\Leftrightarrow cos\left(4x-\dfrac{\pi}{6}\right)=cos\left(\dfrac{\pi}{6}\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}4x-\dfrac{\pi}{6}=\dfrac{\pi}{6}+k2\pi\\4x-\dfrac{\pi}{6}=-\dfrac{\pi}{6}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{12}+\dfrac{k\pi}{2}\\x=\dfrac{k\pi}{2}\end{matrix}\right.\) (\(k\in Z\))