a.
\(sinx+2sinx.cosx=cosx+2cos^2x\)
\(\Leftrightarrow sinx\left(1+2cosx\right)=cosx\left(1+2cosx\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx=cosx\\1+2cosx=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}tanx=1\\cosx=-\frac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{4}+k\pi\\x=\pm\frac{2\pi}{3}+k2\pi\end{matrix}\right.\)
b.
\(\Leftrightarrow\sqrt{2}sin\left(x-\frac{\pi}{4}\right)+\sqrt{2}sin\left(2x-\frac{\pi}{4}\right)=0\)
\(\Leftrightarrow sin\left(2x-\frac{\pi}{4}\right)=-sin\left(x-\frac{\pi}{4}\right)\)
\(\Leftrightarrow sin\left(2x-\frac{\pi}{4}\right)=sin\left(\frac{\pi}{4}-x\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}2x-\frac{\pi}{4}=\frac{\pi}{4}-x+k2\pi\\2x-\frac{\pi}{4}=\frac{3\pi}{4}+x+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{6}+\frac{k2\pi}{3}\\x=\pi+k2\pi\end{matrix}\right.\)
c.
\(1-cos^22x-\left(1+cos2x\right)+\frac{3}{4}=0\)
\(\Leftrightarrow-cos^22x-cos2x+\frac{3}{4}=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos2x=\frac{1}{2}\\cos2x=-\frac{3}{2}\left(l\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{6}+k\pi\\x=-\frac{\pi}{6}+k\pi\end{matrix}\right.\)
d.
ĐKXĐ: ...
\(\Leftrightarrow5+cos2x=2cosx\left(3+\frac{2sinx}{cosx}\right)\)
\(\Leftrightarrow5+cos2x=6cosx+4sinx\)
\(\Leftrightarrow5+cos^2x-sin^2x=6cosx+4sinx\)
\(\Leftrightarrow cos^2x-6cosx+5=sin^2x+4sinx\)
\(\Leftrightarrow cos^2x-6cosx+9=sin^2x+4sinx+4\)
\(\Leftrightarrow\left(cosx-3\right)^2=\left(sinx+2\right)^2\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx+2=cosx-3\\sinx+2=3-cosx\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx-sinx=5>\sqrt{2}\left(l\right)\\sinx+cosx=1\end{matrix}\right.\)
\(\Leftrightarrow\sqrt{2}sin\left(x+\frac{\pi}{4}\right)=1\)
\(\Leftrightarrow sin\left(x+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}\)
\(\Leftrightarrow...\)
