1.
\(\Leftrightarrow\frac{1}{2}-\frac{1}{2}cos\left(2x-\frac{\pi}{3}\right)+\frac{1}{2}-\frac{1}{2}cos\left(2x+\frac{\pi}{3}\right)=5cosx-2\)
\(\Leftrightarrow-\frac{1}{2}\left[cos\left(2x-\frac{\pi}{3}\right)+cos\left(2x+\frac{\pi}{3}\right)\right]=5cosx-3\)
\(\Leftrightarrow-cos2x.cos\frac{\pi}{3}=5cosx-3\)
\(\Leftrightarrow-\frac{1}{2}cos2x=5cosx-3\)
\(\Leftrightarrow cos2x+10cosx-6=0\)
\(\Leftrightarrow2cos^2x+10cosx-7=0\)
\(\Leftrightarrow cosx=\frac{\sqrt{39}-5}{2}\)
\(\Rightarrow x=\pm arccos\left(\frac{\sqrt{39}-5}{2}\right)+k2\pi\)
2.
\(\Leftrightarrow4\left(1-cos^2x\right)^2+12cos^2x-7=0\)
\(\Leftrightarrow4cos^4x+4cos^2x-3=0\)
\(\Leftrightarrow\left(2cos^2x-1\right)\left(2cos^2x+3\right)=0\)
\(\Leftrightarrow2cos^2x-1=0\)
\(\Leftrightarrow cos2x=0\)
\(\Leftrightarrow2x=\frac{\pi}{2}+k\pi\)
\(\Rightarrow x=\frac{\pi}{4}+\frac{k\pi}{2}\)
3.
\(\Leftrightarrow\left(sin^2x+cos^2x\right)^2-2sin^2x.cos^2x=\frac{1}{2}\)
\(\Leftrightarrow1-\frac{1}{2}\left(2sinx.cosx\right)^2=\frac{1}{2}\)
\(\Leftrightarrow1-sin^22x=0\)
\(\Leftrightarrow cos^22x=0\)
\(\Leftrightarrow cos2x=0\)
\(\Leftrightarrow2x=\frac{\pi}{2}+k\pi\)
\(\Leftrightarrow x=\frac{\pi}{4}+\frac{k\pi}{2}\)
4.
\(\Leftrightarrow\left(sin^2x+cos^2x\right)^2-2sin^2x.cos^2x=cos2x\)
\(\Leftrightarrow1-\frac{1}{2}sin^22x=cos2x\)
\(\Leftrightarrow1+1-sin^22x=2cos2x\)
\(\Leftrightarrow1+cos^22x=2cos2x\)
\(\Leftrightarrow\left(cos2x-1\right)^2=0\)
\(\Leftrightarrow cos2x=1\)
\(\Leftrightarrow2x=k2\pi\)
\(\Rightarrow x=k\pi\)
