Find the missing value in the inequality below.

Since both $2^{\sin x}, 2^{\cos x} > 0$ , we can apply AM-PM inequality .

$\large \begin{aligned} 2^{\sin x} + 2^{\cos x} & \ge 2\sqrt{2^{\sin x + \cos x}} \\ & \ge 2\sqrt{2^{\sqrt 2 \color{#3D99F6} \sin \left(x + \frac \pi 4\right)}} & \small \color{#3D99F6} \text{Note that }\min \left(\sin \left(x + \frac \pi 4\right) \right) = -1 \\ & \ge 2\sqrt{2^{{\color{#3D99F6}-}\sqrt 2}} = 2^{1-\frac 1{\sqrt 2}} & \small \color{#3D99F6} \text{when } x = \frac {5\pi}4 \text{and equality occurs.} \end{aligned}$

$\implies a = 1-\dfrac 1{\sqrt 2} \approx \boxed{0.293}$