a/
\(\Leftrightarrow3\left(1-sin^22x\right)+4sin2x-4=0\)
\(\Leftrightarrow-3sin^22x+4sin2x-1=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin2x=1\\sin2x=\frac{1}{3}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{4}+k\pi\\x=\frac{1}{2}arcsin\left(\frac{1}{3}\right)+k\pi\\x=\frac{\pi}{2}-\frac{1}{2}arcsin\left(\frac{1}{3}\right)+k\pi\end{matrix}\right.\)
b/
\(\Leftrightarrow\left[{}\begin{matrix}cos2x=-1\\cos2x=\frac{\sqrt{2}}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{2}+k\pi\\x=\frac{\pi}{8}+k\pi\\x=-\frac{\pi}{8}+k\pi\end{matrix}\right.\)
c/
\(\Leftrightarrow2\left(1-cos^2x\right)-\left(2+\sqrt{3}\right)cosx-2-\sqrt{3}=0\)
\(\Leftrightarrow2cos^2x+\left(2+\sqrt{3}\right)cosx+\sqrt{3}=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=-1\\cosx=-\frac{\sqrt{3}}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\pi+k2\pi\\x=\pm\frac{\pi}{6}+k2\pi\\\end{matrix}\right.\)
d/
\(-2\left(1-2sin^22x\right)-2\left(1-\sqrt{3}\right)sin2x-\sqrt{3}+2=0\)
\(\Leftrightarrow4sin^22x-2\left(1-\sqrt{3}\right)sin2x-\sqrt{3}=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx=\frac{1}{2}\\sinx=-\frac{\sqrt{3}}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{6}+k2\pi\\x=\frac{5\pi}{6}+k2\pi\\x=-\frac{\pi}{3}+k2\pi\\x=\frac{4\pi}{3}+k2\pi\end{matrix}\right.\)
f/
\(\Leftrightarrow4\left(1-2sin^2\frac{x}{2}\right)-5sin\frac{x}{2}=1\)
\(\Leftrightarrow8sin^2\frac{x}{2}+5sin\frac{x}{2}-3=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin\frac{x}{2}=-1\\sin\frac{x}{2}=\frac{3}{8}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\pi+k4\pi\\x=2arcsin\left(\frac{3}{8}\right)+k4\pi\\x=2\pi-2arcsin\left(\frac{3}{8}\right)+k4\pi\end{matrix}\right.\)
