1. Given that the function $f(x)$ is a monotonically increasing function defined on $[-4,+\infty)$, to ensure that for all real numbers $x$,
$$
f\left(\cos x-b^{2}\right) \geqslant f\left(\sin ^{2} x-b-3\right)
$$

holds true. Find the range of real numbers $b$.
Answer $: b \in\left[\frac{1}{2}-\sqrt{2}, 1\right]$.