If ⁻¹(x+1)+ ⁻¹ x+ ⁻¹(x-1)= ⁻¹ 3 , then for x < 0 the value of 500 x^4+270 x^2+997=

Step-by-step solution

The equation ⁻¹(x+1) + ⁻¹ x + ⁻¹(x-1) = ⁻¹ 3 can be rearranged as ⁻¹(x+1) + ⁻¹(x-1) = ⁻¹ 3 - ⁻¹ x . Using the sum and difference formulas for inverse tangent, we apply ⁻¹ A + ⁻¹ B = ⁻¹ ( A+B 1-AB ) to the left side and ⁻¹ A - ⁻¹ B = ⁻¹ ( A-B 1+AB ) to the right side, provided the principal values are valid. The left side simplifies to ⁻¹ ( 2x 2-x^2 ) and the right side becomes ⁻¹ ( 3-x 1+3x ) . Equating the arguments gives the rational equation 2x 2-x^2 = 3-x 1+3x . Cross-multiplying and rearranging yields the cubi