Concept:Use inverse trigonometric angle substitution, then apply trigonometric identities and verify the sign of the solution.Explanation:Let sin−1(4x)=α and sin−1(43x)=β.Then sinα=4x and sinβ=43x.Given α+β=−2π, so β=−2π−α.Taking sine on both sides:sinβ=sin(−2π−α)=−cosα.Thus 43x=−cosα, giving cosα=−43x.Now use cos2α=1−sin2α:(−43x)2=1−(4x)2.So 48x2=1−16x2.Hence 64x2=1, giving x=±81.Check the sign: for x=81, the sum is 2π, not −2π.For x=−81, sin−1(−21)+sin−1(−23)=−6π−3π=−2π.So the valid value is x=−81.Answer:x=−81, i.e. option A.