$\begin{array}{l}
- 1 \le \cos \left( {\dfrac{x}{2} - \dfrac{\pi }{4}} \right) \le 1\\
\Rightarrow 0 \le {\cos ^2}\left( {\dfrac{x}{2} - \dfrac{\pi }{4}} \right) \le 1\\
{\cos ^2}\left( {\dfrac{x}{2} - \dfrac{\pi }{4}} \right) = m
\end{array}$
Phương trình có nghiệm khi $0\le m\le 1$
Vậy $m\in [0;1]$