1.
\(\Leftrightarrow\left[{}\begin{matrix}cosx=1\\cosx=\dfrac{2}{3}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=k2\pi\\x=\pm arccos\left(\dfrac{2}{3}\right)+k2\pi\end{matrix}\right.\)
2.
\(1-cos^2\dfrac{x}{2}-2cos\dfrac{x}{2}+2=0\)
\(\Leftrightarrow cos^2\dfrac{x}{2}+2cos\dfrac{x}{2}-3=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos\dfrac{x}{2}=1\\cos\dfrac{x}{2}=-3< -1\left(loại\right)\end{matrix}\right.\)
\(\Leftrightarrow\dfrac{x}{2}=k2\pi\)
\(\Leftrightarrow x=k4\pi\)
3.
\(3\left(1-sin^2x\right)-2sinx+2=0\)
\(\Leftrightarrow-3sin^2x-2sinx+5=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx=1\\sinx=-\dfrac{5}{3}< -1\left(loại\right)\end{matrix}\right.\)
\(\Leftrightarrow x=\dfrac{\pi}{2}+k2\pi\)
4.
\(\left(sin^2x+cos^2x\right)^3-3sin^2x.cos^2x\left(sin^2x+cos^2x\right)=4cos^22x\)
\(\Leftrightarrow1-\dfrac{3}{4}sin^22x=4\left(1-sin^22x\right)\)
\(\Leftrightarrow sin^22x=\dfrac{12}{13}\)
\(\Leftrightarrow\dfrac{1-cos4x}{2}=\dfrac{12}{13}\)
\(\Leftrightarrow cos4x=-\dfrac{11}{13}\)
\(\Leftrightarrow x=\pm\dfrac{1}{4}arccos\left(-\dfrac{11}{13}\right)+\dfrac{k\pi}{2}\)
5.
\(1-cos^2x-\dfrac{1}{4}=cos^4x\)
\(\Leftrightarrow cos^4x+cos^2x-\dfrac{3}{4}=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos^2x=\dfrac{1}{2}\\cos^2x=-\dfrac{3}{2}\left(loại\right)\end{matrix}\right.\)
\(\Leftrightarrow2cos^2x=1\)
\(\Leftrightarrow2cos^2x-1=0\)
\(\Leftrightarrow cos2x=0\)
\(\Leftrightarrow2x=\dfrac{\pi}{2}+k\pi\)
\(\Leftrightarrow x=\dfrac{\pi}{4}+\dfrac{k\pi}{2}\)
6.
\(4sin^4x+12\left(1-sin^2x\right)=7\)
\(\Leftrightarrow4sin^4x-12sin^2x+5=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin^2x=\dfrac{5}{2}>1\left(loại\right)\\sin^2x=\dfrac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow2sin^2x=1\)
\(\Leftrightarrow1-2sin^2x=0\)
\(\Leftrightarrow cos2x=0\)
\(\Leftrightarrow x=\dfrac{\pi}{4}+\dfrac{k\pi}{2}\)
7.
\(tan^2\left(2x-\dfrac{\pi}{4}\right)=3\)
\(\Leftrightarrow\left[{}\begin{matrix}tan\left(2x-\dfrac{\pi}{4}\right)=\sqrt{3}\\tan\left(2x-\dfrac{\pi}{4}\right)=-\sqrt{3}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}2x-\dfrac{\pi}{4}=\dfrac{\pi}{3}+k\pi\\2x-\dfrac{\pi}{4}=-\dfrac{\pi}{3}+k\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}2x=\dfrac{7\pi}{12}+k\pi\\2x=-\dfrac{\pi}{12}+k\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{7\pi}{24}+\dfrac{k\pi}{2}\\x=-\dfrac{\pi}{24}+\dfrac{k\pi}{2}\end{matrix}\right.\)
8.
\(cot^2x+\left(\sqrt{3}-1\right)cotx-\sqrt{3}=0\)
\(\Leftrightarrow cot^2x-cotx+\sqrt{3}cotx-\sqrt{3}=0\)
\(\Leftrightarrow cotx\left(cotx-1\right)+\sqrt{3}\left(cotx-1\right)=0\)
\(\Leftrightarrow\left(cotx-1\right)\left(cotx+\sqrt{3}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cotx=1\\cotx=-\sqrt{3}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{4}+k\pi\\x=-\dfrac{\pi}{6}+k\pi\end{matrix}\right.\)
