\(\begin{array}{l}a)\,\,\cos \left( {x - 1} \right) = \frac{2}{3}\\ \Leftrightarrow \left[ \begin{array}{l}x - 1 = \arccos \frac{2}{3} + k2\pi \\x - 1 = - \arccos \frac{2}{3} + k2\pi \end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l}x = \arccos \frac{2}{3} + 1 + k2\pi \\x = - \arccos \frac{2}{3} + 1 + k2\pi \end{array} \right.\,\,\,\left( {k \in Z} \right)\\b)\,\,\cos 3x = \cos {12^0}\\ \Leftrightarrow \left[ \begin{array}{l}3x = {12^0} + k{360^0}\\3x = - {12^0} + k{360^0}\end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l}x = {4^0} + k{120^0}\\x = - {4^0} + k{120^0}\end{array} \right.\,\,\,\left( {k \in Z} \right)\\c)\,\,\cos \left( {\frac{{3x}}{2} - \frac{\pi }{4}} \right) = - \frac{1}{2}\\ \Leftrightarrow \cos \left( {\frac{{3x}}{2} - \frac{\pi }{4}} \right) = \cos \frac{{2\pi }}{3}\\ \Leftrightarrow \left[ \begin{array}{l}\frac{{3x}}{2} - \frac{\pi }{4} = \frac{{2\pi }}{3} + k2\pi \\\frac{{3x}}{2} - \frac{\pi }{4} = - \frac{{2\pi }}{3} + k2\pi \end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l}\frac{{3x}}{2} = \frac{{11\pi }}{{12}} + k2\pi \\\frac{{3x}}{2} = - \frac{{5\pi }}{{12}} + k2\pi \end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l}x = \frac{{11\pi }}{{18}} + \frac{{4k\pi }}{3}\\x = \frac{{ - 5\pi }}{{18}} + \frac{{4k\pi }}{3}\end{array} \right.\,\,\,\left( {k \in Z} \right)\\d)\,\,{\cos ^2}2x = \frac{1}{4}\\ \Leftrightarrow \left[ \begin{array}{l}\cos 2x = \frac{1}{2} = \cos \frac{\pi }{3}\\\cos 2x = - \frac{1}{2} = \cos \frac{{2\pi }}{3}\end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l}2x = \pm \frac{\pi }{3} + k2\pi \\2x = \pm \frac{{2\pi }}{3} + k2\pi \end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l}x = \pm \frac{\pi }{6} + k\pi \\x = \pm \frac{\pi }{3} + k\pi \end{array} \right.\,\,\,\left( {k \in Z} \right)\end{array}\)
