$\begin{array}{l} - 1 \le \cos \left( {x - \dfrac{\pi }{3}} \right) \le 1\\ \Leftrightarrow 0 \le {\cos ^2}\left( {x - \dfrac{\pi }{3}} \right) \le 1\\ \Rightarrow 0 \le \sqrt 5 {\cos ^2}\left( {x - \dfrac{\pi }{3}} \right) \le \sqrt 5 \\ \Rightarrow 1 \le y \le \sqrt 5 + 1\\ \Rightarrow \left\{ \begin{array}{l} \max y = \sqrt 5 + 1 \Rightarrow \cos \left( {x - \dfrac{\pi }{3}} \right) = 1 \to x - \dfrac{\pi }{3} = k2\pi \\ \min y = 1 \Rightarrow \cos \left( {x - \dfrac{\pi }{3}} \right) = 0 \Rightarrow x - \dfrac{\pi }{3} = \dfrac{\pi }{2} + k\pi \end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l} \max y = \sqrt 5 + 1 \Rightarrow \cos \left( {x - \dfrac{\pi }{3}} \right) = 1 \to x = \dfrac{\pi }{3} + k2\pi \\ \min y = 1 \Rightarrow \cos \left( {x - \dfrac{\pi }{3}} \right) = 0 \Rightarrow x = \dfrac{{5\pi }}{6} + k\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right) \end{array}$
