Evaluate the following: `\ (x+sqrt(x^2-1))^6+(x-sqrt(x^2-1))^6`

Explanation

Let
Then , `(x+ a)^(6) + (x - a)^(6) = 2 {""^(6)C_(0) x^(6-0) a^(0) + ""^(6)C_(2) + x^(6-2) a^(2) + ""^(6)C_(4) x^(6-4) a^(4) + ""^(6)C_(6) x^(6-6) a^(4)}`
`= 2 {x^(6) + 15x^(4) a^(2) + 15x^(2) + 15x^(2) a^(4) + a^(6)} " " ` [from point (2)]
`= 2 {x^(6) 15x^(4) (x^(2) -1) + 15 x^(2) (x^(2) -1)^(2) + (x^(2) -1)^(3)} " " [ because a = sqrt(x^(2) -1)]`
` = 2(32 x^(6) - 48 x^(6) - 48x^(4) + 18x^(2) -1)` .