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