If 2 sin x / 1+sinx+ cos x = k then 1+sin x -cosx /1+sinx = 1) 1/k2) k3)1-k4)1+k

put sinx= 2 sinx/2 cosx/2
1+cosx = 2cos^2 x/2
so the first expression becomes
(4sinx/2cosx/2)/(2cos^2 x/2+ 2sinx/2cosx/2)
on simplifying
(2 sinx/2)/ (cosx/2+sinx/2)= k.......... (1)

now in the second expression
1-cos x= 2sin^2 x/2
putting this in the expression we get
(2sin^2 x/2+2sinx/2cosx/2)/( cos^2 x/2 + sin^2 x/2 + 2sinx/2cosx/2)
now by simplifying this we get
2sinx/2(sinx/2+cosx/2)/(sinx/2+cosx/2)^2
= 2sinx/2/ (cosx/2+sinx/2)............... (2)
on comparing (1) and (2), we get
(1+sinx-cosx)/(1+sinx)= k
so the correct option is (2) K........... [ans]