$\operatorname{Cos}(\alpha \pm \beta )=\operatorname{Cos}\alpha \operatorname{Cos}\beta \mp \operatorname{Sin}\alpha \operatorname{Sin}\beta \Rightarrow \operatorname{Cos}(x\pm \frac{\pi }{4})=\operatorname{Cos}x\operatorname{Cos}\frac{\pi }{4}\mp \operatorname{Sin}x\operatorname{Sin}\frac{\pi }{4}=\frac{\sqrt{2}}{2}(\operatorname{Cos}x\mp \operatorname{Sin}x)$  $\operatorname{Cos}(x+\frac{\pi }{4})\operatorname{Cos}(x-\frac{\pi }{4})=\frac{1}{4}\Rightarrow \frac{\sqrt{2}}{2}(\operatorname{Cos}x-\operatorname{Sin}x)\times \frac{\sqrt{2}}{2}(\operatorname{Cos}x+\operatorname{Sin}x)=\frac{1}{4}\Rightarrow \frac{1}{2}(\underbrace{{{\operatorname{Cos}}^{2}}x+{{\operatorname{Sin}}^{2}}x}_{\operatorname{Cos}2x})=\frac{1}{4}\Rightarrow \operatorname{Cos}2x=\frac{1}{2}=\operatorname{Cos}\frac{\pi }{3}\Rightarrow 2x=2k\pi \pm \frac{\pi }{3}\xrightarrow{\div 2}x=k\pi \pm \frac{\pi }{6}$