b.
\(\Leftrightarrow\dfrac{1}{2}-\dfrac{1}{2}cos4x+\dfrac{1}{2}-\dfrac{1}{2}cos8x-1=0\)
\(\Leftrightarrow cos8x+cos4x=0\)
\(\Leftrightarrow2cos^24x+cos4x-1=0\)
\(\Rightarrow\left[{}\begin{matrix}cos4x=-1\\cos4x=\dfrac{1}{2}\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}4x=\pi+k2\pi\\4x=\pm\dfrac{\pi}{3}+k2\pi\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{4}+\dfrac{k\pi}{2}\\x=\pm\dfrac{\pi}{12}+\dfrac{k\pi}{2}\end{matrix}\right.\)
c.
\(2-3cos^2x=2sinx+3sin^2x\)
\(\Leftrightarrow2=2sinx+3\left(sin^2x+cos^2x\right)\)
\(\Leftrightarrow2=2sinx+3\)
\(\Leftrightarrow sinx=-\dfrac{1}{2}\)
\(\Rightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{6}+k2\pi\\x=\dfrac{7\pi}{6}+k2\pi\end{matrix}\right.\)
d.
\(3sinx+cosx=\sqrt{2}sin\left(2x+\dfrac{\pi}{4}\right)-2\)
\(\Leftrightarrow3sinx+cosx=sin2x+cos2x-2\)
\(\Leftrightarrow3sinx+cosx=2sinx.cosx+2cos^2x-1-2\)
\(\Leftrightarrow2sinx.cosx-3sinx+2cos^2x-cosx-3=0\)
\(\Leftrightarrow sinx\left(2cosx-3\right)+\left(cosx+1\right)\left(2cosx-3\right)=0\)
\(\Leftrightarrow\left(2cosx-3\right)\left(sinx+cosx+1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=\dfrac{3}{2}>1\left(vn\right)\\sinx+cosx+1=0\left(1\right)\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow\sqrt{2}sin\left(x+\dfrac{\pi}{4}\right)+1=0\)
\(\Leftrightarrow sin\left(x+\dfrac{\pi}{4}\right)=-\dfrac{\sqrt{2}}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}x+\dfrac{\pi}{4}=-\dfrac{\pi}{4}+k2\pi\\x+\dfrac{\pi}{4}=\dfrac{5\pi}{4}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{2}+k2\pi\\x=\pi+k2\pi\end{matrix}\right.\)
