But how?

First we divide by $\cos^2 x$ getting:

$I = \displaystyle \int_{0}^{\infty} \frac{x \tan x \sec x + \sec x}{(x \sec x)^2 + 1} dx$

Now, we put $x \sec x = z$ , checking that $dz = (x \tan x \sec x + \sec x)\text{ dx}$ , and also $z = 0$ at $x=0$ , $z= \infty$ at $x = \infty$ .

Hence, we get $I = \displaystyle \int_{0}^{\infty} \frac{dz}{z^2 + 1} = \frac{\pi}{2}$

Hence, we get $345 \tan(\cos (\sin I)) \approx 206.945$ , so answer is $207$