11. Let $f(x)$ be an increasing function defined on $[1,+\infty)$, and the inequality
$$
f\left(k-\cos ^{2} x\right) \leqslant f\left(k^{2}+\sin x\right)
$$

holds for all $x$. Find the range of real numbers $k$.
$$
\begin{array}{l}
\left\{\begin{array}{l}
k-\cos ^{2} x \geqslant 1, \\
k-\cos ^{2} x \leqslant k^{2}+\sin x
\end{array}\right. \\
\Rightarrow\left\{\begin{array}{l}
k \geqslant 1+\cos ^{2} x, \\
k^{2}-k \geqslant \sin ^{2} x-\sin x-1 .
\end{array}\right.
\end{array}
$$

From equation (1), we get $k \geqslant 2$.
Since $\sin ^{2} x-\sin x-1=\left(\sin x-\frac{1}{2}\right)^{2}-\frac{5}{4} \leqslant 1$, therefore, from equation (2) we get
$k^{2}-k-1 \geqslant 0 \Rightarrow k \leqslant \frac{1-\sqrt{5}}{2}$ or $k \geqslant \frac{1+\sqrt{5}}{2}$.
Thus, $k \geqslant 2$.