Consider the following inequality with $d=\sqrt{r^2+a^2}$ $$2r^2t(1-\cos{2\theta})+4a^2t-\frac{\sin{4dt}}{2d}[(d^2+a^2)\cos{2\theta}-r^2]-a(1-\cos{4dt})\sin{2\theta}>0$$ where all variables are real, $t>0,r>0$ and $a\neq0$. In the setting, $a$ is some constant while time $t$ can grow and it is in the 2D plane $r(\cos{\theta},\sin{\theta})$. In the original problem, we have a 2D function $f(r,\theta)$ and we want to prove $\partial_rf$, given above, has a definite sign. Since this is already self-contained, I feel probably digressing to show $f$.
The first two terms linear in $t$, which are positive, presumably dominates when $t$ is large enough. But I notice from very extended numerics that it actually holds in general. So can we prove it in general? Or something simpler: estimating a lower bound $t_c$, which only depends on $a$, to guarantee its positivity.