Rewrite sectan−1v−4v2 as an algebraic expression in v.

The given expression is sec(tan^-1(v) - 4v^2).

Step 1: We know that sec(x) = 1/cos(x). So, we can rewrite the expression as 1/cos(tan^-1(v) - 4v^2).

Step 2: We also know that cos(tan^-1(x)) = 1/sqrt(1+x^2). So, we can substitute this into the expression to get 1/(1/sqrt(1+v^2) - 4v^2).

Step 3: Simplify the expression to get sqrt(1+v^2)/(1 - 4v^2).

So, the algebraic expression in v for sec(tan^-1(v) - 4v^2) is sqrt(1+v^2)/(1 - 4v^2).