Expand `{(a+b)^(4) + (a-b)^(4)}` and use it to evalute

Explanation

We have
`(a+b)^(4) + ( a-b) ^(4)= [.^(4)C_(0)a^(4)+.^(4)C_(1)a^(3)b +.^(4)C_(2)a^(2)b^(2)+.^(4)C_(3)ab^(3) + .^(4)C_(4)b^(4)]`
`+[.^(4)C_(0)a^(4) - .^(4)C_(1)a^(3)b + .^(4)C_(2)a^(2)b^(2)-.^(4)C_(3)ab^(3) +.^(4)C_(4)b^(4)]`
`=2[.^(4)C_(0) a^(4) + .^(4)Ca^(2)b^(2) + .^(4)C_(4)b^(4)]." "`...(i)
Putting `sqrt(1-x^(2))=y` ,we get
`(x^(2) + sqrt(1-x^(2)))^(4)+ ( x^(2)- sqrt(1-x^(2)))^(4)`
`=(x^(2) + y)^(4) + ( x^(2) - y)^(2)`
`=2[.^(4)C_(0)(x^(2))^(4) + .^(4)C_(2)(x^(2))^(2) y^(2)+.^(4)C_(4)y^(4)]`
[taking `a=x^(2) and b= y `in (i) ]
`2[x^(8) + 6x^(4) y^(2) + y^(2)]`
`=2[x^(8) + 6x^(4) (1-x^(2)) + (1-x^(2))^(2) ]" "[:' y^(2) = (1-x^(2))]`
`2[x^(8) + ( 6x^(4) -6x^(6))+ ( 1-2x^(2) + x^(4))]`
`=2[x^(8)-6x^(6) + 7x^(4) - 2x^(2) + 1]`
`2x^(8) - 12 x^(6) + 14x^(4) - 4x^(2) + 2.`