1.
\(\Leftrightarrow2sin\frac{x}{2}cos\frac{x}{2}+\sqrt{3}cos\frac{x}{2}=0\)
\(\Leftrightarrow cos\frac{x}{2}\left(2sin\frac{x}{2}+\sqrt{3}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos\frac{x}{2}=0\\sin\frac{x}{2}=-\frac{\sqrt{3}}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\frac{x}{2}=\frac{\pi}{2}+k\pi\\\frac{x}{2}=-\frac{\pi}{3}+k2\pi\\\frac{x}{2}=\frac{4\pi}{3}+k2\pi\end{matrix}\right.\) \(\Leftrightarrow...\)
2.
\(\Leftrightarrow cosx=2cos^2\left(\frac{x}{2}-\frac{\pi}{6}\right)-1\)
\(\Leftrightarrow cosx=cos\left(x-\frac{\pi}{3}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}x=x-\frac{\pi}{3}+k2\pi\left(vn\right)\\x=\frac{\pi}{3}-x+k2\pi\end{matrix}\right.\)
\(\Rightarrow x=\frac{\pi}{6}+k\pi\)
3.
\(\Leftrightarrow\frac{1}{2}sinx-\frac{\sqrt{3}}{2}cosx=0\)
\(\Leftrightarrow sin\left(x-\frac{\pi}{3}\right)=0\)
\(\Leftrightarrow x-\frac{\pi}{3}=k\pi\)
\(\Leftrightarrow...\)
4.
\(1+\frac{1}{2}sin6x=sin^2x+cos^2x+2sinx.cosx\)
\(\Leftrightarrow\frac{1}{2}sin6x=sin2x\)
\(\Leftrightarrow sin6x-2sin2x=0\)
\(\Leftrightarrow3sin2x-4sin^32x-2sin2x=0\)
\(\Leftrightarrow sin2x-4sin^32x=0\)
\(\Leftrightarrow sin2x\left(1-4sin^22x\right)=0\)
\(\Leftrightarrow sin2x\left(2cos2x-1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin2x=0\\cos2x=\frac{1}{2}\end{matrix}\right.\) \(\Leftrightarrow...\)
