The most general values of x for which $\sin \mathrm{x}+\cos \mathrm{x}=\min \left\{1, {\mathrm{a}}^2-4\mathrm{a}+6\right\}, \mathrm{a}\in \mathrm{R}$ are given by

$$\begin{gathered} \sin \mathrm{x}+\cos \mathrm{x}=\mathrm{Min} \mathrm{a}\in \mathrm{R}\left\{1, {\left(\mathrm{a}-2\right)}^2+2\right\} \\ \sin \mathrm{x}+\cos \mathrm{x}=1 \\ \frac{1}{\sqrt{2}}\sin \mathrm{x}+\frac{1}{\sqrt{2}}\cos \mathrm{x}=\frac{1}{\sqrt{2}} \\ \sin \left(\mathrm{x}+\frac{\mathrm{\pi }}{4}\right) =\sin \frac{\mathrm{\pi }}{4} \\ \mathrm{x}+\frac{\mathrm{\pi }}{4}=\mathrm{n\pi }+{(-1)}^{\mathrm{n}}\frac{\mathrm{\pi }}{4} \mathrm{x}=\mathrm{n\pi }+{(-1)}^{\mathrm{n}}\frac{\mathrm{\pi }}{4}-\frac{\mathrm{\pi }}{4} \end{gathered}$$