Rewrite $\cos \left(2 \sin ^{-1} 4 v\right)$ as an algebraic expression in $v$. \[ \cos \left(2 \sin ^{-1} 4 v\right)=\square \]

Step 1 ：Given the expression \(\cos \left(2 \sin ^{-1} 4 v\right)\), we need to rewrite it as an algebraic expression in terms of \(v\).

Step 2 ：We know that the double angle formula for cosine is \(\cos(2A) = 1 - 2\sin^2(A)\), where \(A = \sin^{-1}(4v)\).

Step 3 ：We can substitute \(A\) into the equation, but first we need to express \(\sin^2(A)\) in terms of \(v\).

Step 4 ：Since \(\sin(A) = 4v\), we have \(\sin^2(A) = (4v)^2 = 16v^2\).

Step 5 ：Substituting this into the equation, we get \(\cos(2A) = 1 - 2*(16v^2) = 1 - 32v^2\).

Step 6 ：Thus, the algebraic expression for \(\cos \left(2 \sin ^{-1} 4 v\right)\) in terms of \(v\) is \(1 - 32v^2\).

Step 7 ：Final Answer: \(\boxed{1 - 32v^2}\)