a.
\(\Leftrightarrow\pi cos2x=\frac{\pi}{2}+k2\pi\)
\(\Leftrightarrow cos2x=\frac{1}{2}+2k\)
Do \(-1\le cos2x\le1\Rightarrow-1\le\frac{1}{2}+2k\le1\)
\(\Rightarrow k=0\)
\(\Rightarrow cos2x=\frac{1}{2}\)
\(\Rightarrow\left[{}\begin{matrix}x=\frac{\pi}{3}+k2\pi\\x=-\frac{\pi}{3}+k2\pi\end{matrix}\right.\)
b.
\(\Leftrightarrow cos4x=1\)
\(\Leftrightarrow4x=k2\pi\)
\(\Leftrightarrow x=\frac{k\pi}{2}\)
c.
ĐKXĐ: \(cosx\ne1\)
\(\Leftrightarrow cos2x-1=1-cosx\)
\(\Leftrightarrow2cos^2x-1-1=1-cosx\)
\(\Leftrightarrow2cos^2x+cosx-3=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=1\left(l\right)\\cosx=-\frac{3}{2}< -1\left(l\right)\end{matrix}\right.\)
Vậy pt đã cho vô nghiệm
d.
ĐKXĐ: \(\left\{{}\begin{matrix}cosx\ne0\\tanx\ne1\end{matrix}\right.\)
\(\Leftrightarrow cos2x=tanx-1\)
\(\Leftrightarrow cos^2x-sin^2x=\frac{sinx}{cosx}-1\)
\(\Leftrightarrow\left(cosx-sinx\right)\left(cosx+sinx\right)=\frac{cosx-sinx}{-cosx}\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx-sinx=0\Leftrightarrow tanx=1\left(l\right)\\cosx+sinx=-\frac{1}{cosx}\left(1\right)\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow cos^2x+sinx.cosx=-1\)
\(\Leftrightarrow\frac{1}{2}+\frac{1}{2}cos2x+\frac{1}{2}sin2x=-1\)
\(\Leftrightarrow cos2x+sin2x=-3\)
Do \(\left\{{}\begin{matrix}cos2x\ge-1\\sin2x\ge-1\end{matrix}\right.\) \(\Rightarrow cos2x+sin2x\ge-2>-3\)
\(\Rightarrow\left(1\right)\) vô nghiệm
Vậy pt đã cho vô nghiệm
