3. If the inequality $a(4-\sin x)^{4}-3+\cos ^{2} x+a>0$ holds for any $x \in \mathbf{R}$, then the range of the real number $a$ is $\qquad$
3. $\left(\frac{3}{82},+\infty\right)$.

When $x=\frac{\pi}{2}$, the original inequality holds, yielding $a>\frac{3}{82}$.
When $a>\frac{3}{82}$, for $x \in \mathbf{R}$, we have
$$
a(4-\sin x)^{4}-3+\cos ^{2} x+a \geqslant a \cdot 3^{4}-3+0+a=82 a-3>0 .
$$

Therefore, the range of real numbers $a$ is $\left(\frac{3}{82},+\infty\right)$.