a.
\(cos2x=sinx\)
\(\Leftrightarrow cos2x=cos\left(\frac{\pi}{2}-x\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}2x=\frac{\pi}{2}-x+k2\pi\\2x=x-\frac{\pi}{2}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{6}+\frac{k2\pi}{3}\\x=-\frac{\pi}{2}+k2\pi\end{matrix}\right.\)
b.
\(\Leftrightarrow2sin\frac{x}{2}cos\frac{x}{2}+\sqrt{3}sin\frac{x}{2}=0\)
\(\Leftrightarrow sin\frac{x}{2}\left(2cos\frac{x}{2}+\sqrt{3}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin\frac{x}{2}=0\\cos\frac{x}{2}=-\frac{\sqrt{3}}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\frac{x}{2}=k\pi\\\frac{x}{2}=\frac{5\pi}{6}+k2\pi\\\frac{x}{2}=-\frac{5\pi}{6}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=k2\pi\\x=\frac{5\pi}{3}+k4\pi\\x=-\frac{5\pi}{3}+k4\pi\end{matrix}\right.\)
c.
\(\Leftrightarrow2sin\frac{x}{2}cos\frac{x}{2}-\sqrt{3}cos\frac{x}{2}=0\)
\(\Leftrightarrow cos\frac{x}{2}\left(2sin\frac{x}{2}-\sqrt{3}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos\frac{x}{2}=0\\sin\frac{x}{2}=\frac{\sqrt{3}}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\frac{x}{2}=\frac{\pi}{2}+k\pi\\\frac{x}{2}=\frac{\pi}{3}+k2\pi\\\frac{x}{2}=\frac{2\pi}{3}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\pi+k2\pi\\x=\frac{2\pi}{3}+k4\pi\\x=\frac{4\pi}{3}+k4\pi\end{matrix}\right.\)
d.
ĐKXĐ: ...
\(\Leftrightarrow tan\left(3x-\frac{\pi}{5}\right)=tan\left(\frac{\pi}{2}-x\right)\)
\(\Leftrightarrow3x-\frac{\pi}{5}=\frac{\pi}{2}-x+k\pi\)
\(\Leftrightarrow x=\frac{7\pi}{40}+\frac{k\pi}{4}\)
e.
ĐKXĐ: \(\left\{{}\begin{matrix}cos3x\ne0\\cosx\ne0\end{matrix}\right.\) \(\Leftrightarrow x\ne\frac{\pi}{6}+\frac{k\pi}{3}\)
\(\frac{sin3x.sinx}{cos3x.cosx}=1\)
\(\Leftrightarrow cos3x.cosx=sin3x.sinx\)
\(\Leftrightarrow cos3x.cosx-sin3x.sinx=0\)
\(\Leftrightarrow cos4x=0\)
\(\Leftrightarrow x=\frac{\pi}{8}+\frac{k\pi}{4}\)