1.
\(\Leftrightarrow\frac{\pi}{3}cosx-\frac{8\pi}{3}=k\pi\)
\(\Leftrightarrow cosx=8+3k\)
Do \(-1\le cosx\le1\Rightarrow-1\le8+3k\le1\)
\(\Rightarrow-3\le k\le-\frac{7}{3}\) \(\Rightarrow k=-3\)
\(\Rightarrow cosx=-1\Rightarrow x=\pi+k2\pi\)
2.
\(\Leftrightarrow\frac{\pi}{3}cos2\pi x=\frac{\pi}{6}+k\pi\)
\(\Leftrightarrow cos2\pi x=\frac{1}{2}+3k\)
Do \(-1\le2\pi x\le1\Rightarrow-1\le\frac{1}{2}+3k\le1\)
\(\Rightarrow-\frac{1}{2}\le k\le\frac{1}{6}\Rightarrow k=0\)
\(\Rightarrow cos2\pi x=\frac{1}{2}\Rightarrow\left[{}\begin{matrix}2\pi x=\frac{\pi}{3}+k2\pi\\2\pi x=-\frac{\pi}{3}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{1}{6}+k\\x=-\frac{1}{6}+k\end{matrix}\right.\)
