The minimum value of $a^{{\mathrm{Cos}}^{2}x}+a^{{\sin }^{2}x}\mathrm{\forall }(a>0)$

$$\begin{gathered} \boxed{\mathrm{A}.\mathrm{M}\ge \mathrm{G}.\mathrm{M}} \\ \Rightarrow \frac{{\mathrm{a}}^{{\cos }^2\mathrm{x}}+{\mathrm{a}}^{{\sin }^2\mathrm{x}}}{2}\ge \sqrt{{\mathrm{a}}^{{\cos }^2\mathrm{x}}{\mathrm{a}}^{{\sin }^2\mathrm{x}}} \\ \Rightarrow {\mathrm{a}}^{{\cos }^2\mathrm{x}}+{\mathrm{a}}^{{\sin }^2\mathrm{x}}\ge 2\sqrt{{\mathrm{a}}^{{\cos }^2\mathrm{x}+{\sin }^2\mathrm{x}}} \\ \Rightarrow {\mathrm{a}}^{{\cos }^2\mathrm{x}}+{\mathrm{a}}^{{\sin }^2\mathrm{x}}\ge 2\sqrt{\mathrm{a}} \\ \boxed{\mathrm{Minimum} \mathrm{value} \mathrm{of} {\mathrm{a}}^{{\cos }^2\mathrm{x}}+{\mathrm{a}}^{{\sin }^2\mathrm{x}} \mathrm{is} 2\sqrt{\mathrm{a}}} \end{gathered}$$