To simplify the given expression, we can rewrite sin(x/2 - π/6) in terms of sinx and cosx.

sin(x/2 - π/6) = sin(x/2)cos(π/6) - cos(x/2)sin(π/6)
= sin(x/2)(√3/2) - cos(x/2)(1/2)
= (√3/2)sin(x/2) - (1/2)cos(x/2)

Now our expression becomes:

cosx + √3sinx = (√3/2)sin(x/2) - (1/2)cos(x/2)

Using sum-to-product formulas and double angle formulas, we can further simplify the expression:

cosx + √3sinx = (√3/2)(2sin(x/2)cos(x/2)) - (1/2)(2cos^2(x/2) - 1)
= √3sinx + √3cos(x) - cos^2(x/2) + 1/2

Solving further, we get:

cos^2(x/2) - cosx + sinx - (√3/2)sin(x) + 1/2 = 0

This is the simplified form of the given expression.