In answering Average angle between two randomly chosen vectors in a unit square, I noticed that the average angle formed by two vectors uniformly picked in the unit square, $\frac\pi4+\log2-1\approx0.4785$, and the average angle formed by two vectors uniformly picked in the first quadrant of the unit disk, $\frac\pi6\approx0.5236$, add up to just a bit more than $1$ (about $1.0021$), that is, that
$$ \frac\pi4+\frac\pi6+\log2\gtrsim2\;. $$
I thought it would be interesting to try to prove this with as little numerics as possible. Of course you can use sufficiently good rational approximations for $\pi$ and $\log2$, like I did to answer Prove that $e^\pi+\frac{1}{\pi} < \pi^e+1$, but I’d prefer a proof that shows directly that $\frac\pi4+\frac\pi6+\log2-2$ is a positive quantity, e.g. the integral over a positive function, as in Is there an integral that proves $\pi > 333/106$?.