Let $y=\sin x$ and $z=\cos x$.
Then we want to know if the maximum value of $\cos(\cos x)+\cos(\sin x)=\cos y+\cos z$ is less than $\frac\pi 2$ with the boundary condition $y^2+z^2=1$.
We can find the extrema with Lagrange multipliers.
Define $\Lambda(y,z,\lambda)=\cos y+\cos z-\lambda(y^2+z^2-1). \tag 1$
The extrema occur for $(y,z)$ with:
$$\begin{cases}\Lambda_y=-\sin y -2\lambda y = 0\\
\Lambda_z=-\sin z -2\lambda z = 0 \\
\Lambda_\lambda = -(y^2+z^2-1) = 0 \end{cases} \implies
\begin{cases}-2\lambda = \frac{\sin y}{y} \lor y=0 \\
-2\lambda = \frac{\sin z}{z} \lor z = 0 \\
y^2+z^2=1\end{cases} \implies
\begin{cases}\frac{\sin y}{y} = \frac{\sin z}{z} \lor y=0 \lor z=0 \\
y^2+z^2=1\end{cases} \tag 2
$$
So $(y,z)$ is on the unit circle and $-1\le y,z \le +1$.
Note that $\frac{\sin t}{t}$ is an even function that is bijective on the interval $(0,1]$. Therefore $\frac{\sin y}{y} = \frac{\sin z}{z} \implies y=\pm z$.
It follows that the extrema are when $(y,z)$ is $\left(\pm \frac 1{\sqrt 2},\pm \frac 1{\sqrt 2}\right)$ and/or $(0,\,\pm 1)$ and/or $(\pm 1,\,0)$.
Thus: $\cos(\sin x)+\cos(\cos x) \le \max\left[ \cos\left(\pm \frac 1{\sqrt 2}\right) + \cos\left(\pm \frac 1{\sqrt 2}\right),\,\cos(0) + \cos(\pm 1)\right]
=\max\left[2\cos\left(\frac 1{\sqrt 2}\right),\, 1+\cos 1\right] \tag 3$
From the Taylor expansion we get that $\cos x \le 1-\frac {x^2}2+\frac{x^4}{4!}$.
So $2\cos\left(\frac 1{\sqrt 2}\right) \le 2\left(1-\frac {1/2}2 + \frac {(1/2)^2}{24}\right)=1.5+\frac 1{48} \le 1.5 + \frac 1{20} = 1.55$ and $1+\cos 1 \le 1+ \left(1-\frac 12 + \frac 1{24}\right) \le 1.5 + \frac 1{20}=1.55$.
Finally we have that $\frac\pi 2 > \frac{3.14}{2} = 1.57$, which completes the proof since $1.55 < 1.57$.