All right, let's proceed with this task with geometrical ideas. We know that the formula for the tangent of an angle in a right triangle is opposite over adjacent sides. The arcsine function, asin(x), gives us an angle whose sine is x. In this case, we have asin(x^2), so we are looking for an angle whose sine is x^2.
1. Let's consider a right triangle where the side opposite the angle A is represented by x^2 (given that sin(A) = x^2) and the hypotenuse is 1 (as sin(A) = Opposite/Hypotenuse).
2. The adjacent side of this right triangle would be the square root of [1 - (opposite side)^2] applying the Pythagorean theorem, hence sqrt(1 - (x^2)^2) = sqrt(1 - x^4).
3. The tan of A would then be (opposite side) / (adjacent side) = x^2 / sqrt(1 - x^4).
So, the simplified form of the expression tan(asin(x^2)) is x^2 / sqrt(1 - x^4).