a/
\(\Leftrightarrow cos\frac{4x}{3}=\frac{cos2x+1}{2}\)
Đặt \(\frac{2x}{3}=a\Rightarrow2x=3a\)
Pt trở thành:
\(cos2a=\frac{cos3a+1}{2}\)
\(\Leftrightarrow2\left(2cos^2a-1\right)=4cos^3a-3cosa+1\)
\(\Leftrightarrow4cos^3a-4cos^2a-3cosa+3=0\)
\(\Leftrightarrow\left(cosa-1\right)\left(4cos^2a-3\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}cosa=1\\cosa=\frac{\sqrt{3}}{2}\\cosa=-\frac{\sqrt{3}}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}cos\left(\frac{2x}{3}\right)=1\\cos\left(\frac{2x}{3}\right)=\frac{\sqrt{3}}{2}\\cos\left(\frac{2x}{3}\right)=-\frac{\sqrt{3}}{2}\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}\frac{2x}{3}=k2\pi\\\frac{2x}{3}=\pm\frac{\pi}{6}+k2\pi\\\frac{2x}{3}=\pm\frac{7\pi}{6}+k2\pi\end{matrix}\right.\) \(\Rightarrow x=...\)
b/
Đặt \(\frac{2x}{3}=a\)
\(\Rightarrow cos4a=cos^2a\)
\(\Leftrightarrow2cos^22a-1=\frac{1+cos2a}{2}\)
\(\Leftrightarrow4cos^22a-cos2a-3=0\)
\(\Rightarrow\left[{}\begin{matrix}cos2a=1\\cos2a=-\frac{3}{4}\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}cos\left(\frac{4x}{3}\right)=1\\cos\left(\frac{4x}{3}\right)=-\frac{3}{4}\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}\frac{4x}{3}=k2\pi\\\frac{4x}{3}=\pm arccos\left(-\frac{3}{4}\right)+k2\pi\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=\frac{k3\pi}{2}\\x=\pm\frac{3}{4}arccos\left(-\frac{3}{4}\right)+\frac{k3\pi}{2}\end{matrix}\right.\)
c/
\(\Leftrightarrow cos\frac{6x}{5}+2=3cos\frac{4x}{5}\)
Đặt \(\frac{2x}{5}=a\)
\(\Rightarrow cos3a+2=3cos2a\)
\(\Leftrightarrow4cos^3a-3cosa+2=6cos^2a-3\)
\(\Leftrightarrow4cos^3a-6cos^2a-3cosa+5=0\)
\(\Leftrightarrow\left(cosa-1\right)\left(4cos^2a-2cosa-5\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}cosa=1\\cosa=\frac{1+\sqrt{21}}{4}>1\left(l\right)\\cosa=\frac{1-\sqrt{21}}{4}\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}cos\left(\frac{2x}{5}\right)=1\\cos\left(\frac{2x}{5}\right)=\frac{1-\sqrt{21}}{4}\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}\frac{2x}{5}=k2\pi\\\frac{2x}{5}=\pm arccos\left(\frac{1-\sqrt{21}}{4}\right)+k2\pi\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=k5\pi\\x=\pm\frac{5}{2}arccos\left(\frac{1-\sqrt{21}}{4}\right)+k5\pi\end{matrix}\right.\)
d/
ĐKXĐ: ...
\(\Leftrightarrow cos^2x+\frac{1}{cos^2x}+2=2\left(cosx+\frac{1}{cosx}\right)\)
\(\Leftrightarrow\left(cosx+\frac{1}{cosx}\right)^2=2\left(cox+\frac{1}{cosx}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx+\frac{1}{cosx}=0\\cosx+\frac{1}{cosx}=2\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}cos^2x+1=0\left(vn\right)\\cos^2x-2cosx+1=0\end{matrix}\right.\)
\(\Rightarrow cosx=1\)
\(\Rightarrow x=k2\pi\)
