a, \(sin^2x-4sinx+3=0\)
\(\Leftrightarrow\left(sinx-1\right)\left(sinx-3\right)=0\)
\(\Leftrightarrow sinx=1\)
\(\Leftrightarrow x=\dfrac{\pi}{2}+k2\pi\)
b, \(2cos^2-cosx-1=0\)
\(\Leftrightarrow\left(cosx-1\right)\left(2cosx+1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=1\\cosx=-\dfrac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=k2\pi\\x=\pm\dfrac{2\pi}{3}+k2\pi\end{matrix}\right.\)
c, \(3sin^2x-2cosx+2=0\)
\(\Leftrightarrow3-3sin^2x+2cosx-5=0\)
\(\Leftrightarrow3cos^2x+2cosx-5=0\)
\(\Leftrightarrow\left(cosx-1\right)\left(3cosx+5\right)=0\)
\(\Leftrightarrow cosx=1\)
\(\Leftrightarrow x=k2\pi\)
d, \(3cosx+cos2x-cos3x+1=2sinx.sin2x\)
\(\Leftrightarrow3cosx+2cos^2x-4cos^3x+3cosx=cosx-cos3x\)
\(\Leftrightarrow2cos^2x-4cos^3x+8cosx=0\)
\(\Leftrightarrow2cos^3x-cos^2x-8cosx=0\)
\(\Leftrightarrow cosx=0\)
\(\Leftrightarrow x=\dfrac{\pi}{2}+k\pi\)
e, ĐK: \(x\ne\dfrac{\pi}{2}+k\pi\)
\(tan^2x+\left(\sqrt{3}+1\right)tanx-\sqrt{3}=0\)
\(\Leftrightarrow tanx=\dfrac{-\sqrt{3}-1\pm\sqrt{4+6\sqrt{3}}}{2}\)
\(\Leftrightarrow x=arctan\dfrac{-\sqrt{3}-1\pm\sqrt{4+6\sqrt{3}}}{2}+k\pi\)
f, ĐK: \(x\ne k\pi\)
\(\dfrac{\sqrt{3}}{sin^2x}=3cotx+\sqrt{3}\)
\(\Leftrightarrow\sqrt{3}\left(\dfrac{1}{sin^2x}-1\right)-3cotx=0\)
\(\Leftrightarrow\sqrt{3}cot^2x-3cotx=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cotx=0\\cotx=\sqrt{3}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{4}+k\pi\\x=\dfrac{\pi}{6}+k\pi\end{matrix}\right.\)
