\(\begin{array}{l}
a)2{\cos ^2}2x + \cos 2x = 2{\sin ^2}x + \frac{1}{2}\\
\Rightarrow 2{\cos ^2}2x + \cos 2x = 1 - \cos 2x + \frac{1}{2}\\
\Rightarrow 2{\cos ^2}2x + 2\cos 2x - \frac{3}{2} = 0\\
\Rightarrow \left[ \begin{array}{l}
\cos x = \frac{1}{2}\\
\cos x = \frac{{ - 3}}{2} < - 1(l)
\end{array} \right.\\
\Rightarrow x = \pm \frac{\pi }{3} + k2\pi (k \in )
\end{array}\)