Substituting (√3)/2 into √(1±x), for x, may not look like it has much potential to be simplified, but it can be simplified quite nicely.
This should make plugging in for x much easier.
Let a = 1+x = 1+sqrt(3)/2 and b = 1-x = 1-sqrt(3)/2 ; then expression becomes:
[a(1 - sqrt(b)) + b(1 + sqrt(a))] / [(1 + sqrt(a))(1 - sqrt(b))]
= [a(1 - sqrt(b)) + b(1 + sqrt(a))] / [1 + sqrt(a) - sqrt(b) - sqrt(ab)]
Since sqrt(ab) = 1/2, above simplifies to:
2[a(1 - sqrt(b)) + b(1 + sqrt(a))] / [1 + 2(sqrt(a) - sqrt(b))]
Substitute a = 1+sqrt(3)/2 and b = 1-sqrt(3)/2 and you'll get a nice 1 as answer