Question

Evaluate: `(x^2 - sqrt(1 - x^2))^4 + (x^2 + sqrt(1 - x^2))^4`

Answer 1

Putting `sqrt(1 - x^2)` = y, we get

The given expression = (x² – y)⁴ + (x² + y)⁴

= 2[x⁸ + ⁴C₂ x⁴ y² + ⁴C₄ y⁴]

= `2  x^8 + (4 xx 3)/(2 xx 1) x^4 * (1 - x^2) + (1 - x^2)^2`

= 2[x⁸ + 6x⁴ (1 – x²) + (1 – 2x² + x 4 ]

= 2x⁸ – 12x⁶ + 14x⁴ – 4x² + 2