f, \(3sin^2x-cosx+2cos2x-3=0\)
\(\Leftrightarrow3-3cos^2x-cosx+2\left(2cos^2x-1\right)-3=0\)
\(\Leftrightarrow cos^2x-cosx-2=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=-1\\cosx=2\left(l\right)\end{matrix}\right.\)
\(\Leftrightarrow x=\pi+k2\pi\)
h, \(cos^2x+cos^22x+cos^23x+cos^24x=2\)
\(\Leftrightarrow2cos^2x+2cos^22x+2cos^23x+2cos^24x=4\)
\(\Leftrightarrow cos2x+cos4x+cos6x+cos8x=0\)
\(\Leftrightarrow2cos5x.cos3x+2cos5x.cosx=0\)
\(\Leftrightarrow cos5x\left(cos3x+cosx\right)=0\)
\(\Leftrightarrow2cos5x.cos2x.cosx=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos5x=0\\cos2x=0\\cosx=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}5x=\dfrac{\pi}{2}+k\pi\\2x=\dfrac{\pi}{2}+k\pi\\x=\dfrac{\pi}{2}+k\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{10}+\dfrac{k\pi}{5}\\x=\dfrac{\pi}{4}+\dfrac{k\pi}{2}\\x=\dfrac{\pi}{2}+k\pi\end{matrix}\right.\)
g, \(cos^4x-sin^4x=2cosx-1\)
\(\Leftrightarrow\left(cos^2x-sin^2x\right)\left(cos^2x+sin^2x\right)=2cosx-1\)
\(\Leftrightarrow cos2x-2cosx+1=0\)
\(\Leftrightarrow2cos^2x-2cosx=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=0\\cosx=1\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{2}+k\pi\\x=k2\pi\end{matrix}\right.\)
i, \(sin2x.sin6x=sin7x.sinx\)
\(\Leftrightarrow\dfrac{1}{2}\left(cos4x-cos8x\right)=\dfrac{1}{2}\left(cos6x-cos8x\right)\)
\(\Leftrightarrow cos4x=cos6x\)
\(\Leftrightarrow4x=\pm6x+k2\pi\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-k\pi\\x=\dfrac{k\pi}{5}\end{matrix}\right.\)
j, \(\left(sin\dfrac{x}{2}+cos\dfrac{x}{2}\right)^2+\sqrt{3}cosx=2\)
\(\Leftrightarrow1+sinx+\sqrt{3}cosx=2\)
\(\Leftrightarrow sinx+\sqrt{3}cosx=1\)
\(\Leftrightarrow\dfrac{1}{2}sinx+\dfrac{\sqrt{3}}{2}cosx=\dfrac{1}{2}\)
\(\Leftrightarrow sin\left(x+\dfrac{\pi}{3}\right)=\dfrac{1}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}x+\dfrac{\pi}{3}=\dfrac{\pi}{6}+k2\pi\\x+\dfrac{\pi}{3}=\dfrac{5\pi}{6}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{6}+k2\pi\\x=\dfrac{\pi}{2}+k2\pi\end{matrix}\right.\)
