Rewrite $\sec ({\sin }^{-1}4w)$ as an algebraic expression in $w$.

Step 1 ：Rewrite $\sec ({\sin }^{-1}4w)$ as an algebraic expression in $w$.

Step 2 ：The secant function is the reciprocal of the cosine function. So, $\sec (x)=\frac{1}{\cos (x)}$.

Step 3 ：The inverse sine function, ${\sin }^{-1}(x)$, gives the angle whose sine is $x$. So, ${\sin }^{-1}(4w)$ gives the angle whose sine is $4w$.

Step 4 ：We can use the Pythagorean identity, ${\sin }^2(x)+{\cos }^2(x)=1$, to express cosine in terms of sine: $\cos (x)=\sqrt{1-{\sin }^2(x)}$.

Step 5 ：So, $\sec ({\sin }^{-1}(4w))=\frac{1}{\cos ({\sin }^{-1}(4w))}=\frac{1}{\sqrt{1-{\sin }^2({\sin }^{-1}(4w))}}=\frac{1}{\sqrt{1-(4w{)}^2}}$.

Step 6 ：However, we need to consider the domain of the inverse sine function, which is $-1\le x\le 1$. So, $-1\le 4w\le 1$, which means $-1/4\le w\le 1/4$.

Step 7 ：Final Answer: $\boxed{{\displaystyle \frac{1}{\sqrt{1-16w^2}}}}$, for $-1/4\le w\le 1/4$.