√(1+sinx)-1
=√[sin²(x/2)+cos²(x/2)+2sin(x/2)cos(x/2)] - 1
=√[sin(x/2)+cos(x/2)]² - 1
=|sin(x/2)+cos(x/2)| - 1
=√(2) |sin(x/2+π/4)| - 1