Let x × y = x 2 + y 3 and x × 1 × 1 = x × 1 × 1 . Then a value of 2 sin - 1 x 4 + x 2 - 2 x 4 + x 2 + 2 is

Step-by-step solution

Given x × y = x 2 + y 3 Now x × 1 ×   1 = x × 1 × 1 x 2 + 1 3 × 1 = x × 1 2 + 1 3 x 2 + 1 2 + 1 3 = x 2 + 2 3 x 4 + 1 + 2 x 2 + 1 = x 2 + 8 x 4 + x 2 − 6 = 0 Let x 2 = t t 2 + t - 6 = 0 t + 3 t − 2 = 0 t ≠ − 3 so t = 2 or x 2 = 2 Now 2 sin - 1 x 4 + x 2 - 2 x 4 + x 2 + 2 = 2 sin - 1 4 + 2 - 2 4 + 2 + 2 = 2 sin - 1 1 2 = π 6 × 2 = π 3