Question 16100
|z1+z2|=|z1|+|z2| 
(|z1+z2|) ^2 = (|z1|+|z2| )^2 = |z1|^2+|z2|^2+2|z1||z2|

let z1=x1+iy1
    z2=x2+iy2
hence
 ( |x1+iy1+x2+iy2| )^2 = ( |x1+iy1| )^2+( |x2+iy2| )^2 +2( |x1+iy1| )( |x2+iy2| )
   ( |(x1+x2)+i(y1+y2)| ) ^2 =  (x1^2+y1^2) +(x2^2+y2^2)+2 {{{sqrt((x1^2+y1^2)(x2^2+y2^2))}}}
   (x1+x2)^2+(y1+y2)^2 =  (x1^2+y1^2) +(x2^2+y2^2)+2 {{{sqrt((x1^2+y1^2)(x2^2+y2^2))}}}
x1^2+x2^2+2x1*x2+y1^2+y2^2+2y1*y2 = (x1^2+y1^2) +(x2^2+y2^2)+2 {{{sqrt((x1^2+y1^2)(x2^2+y2^2))}}}
2(x1*x2+y1*y2) = 2 {{{sqrt((x1^2+y1^2)(x2^2+y2^2))}}}
cancelling 2 on either side and squaring both sides 
(x1x2+y1y2)^2 = (x1^2+y1^2)(x2^2+y2^2)
(x1^2)(x2^2)+(y1^2)(y2^2) +2x1*x2*y1*y2 = (x1^2)(x2^2)+(y1^2)(y2^2)+(x1^2)(y2^2)+(x2^2)(y1^2)
(x1^2)(y2^2)+(x2^2)(y1^2) - 2x1*x2*y1*y2 = 0
(x1y2-x2y1)^2 = 0
x1y2 - x2y1 = 0
x1y2 = x2y1
y1/x1 = y2/x2
Tan (y1/x1) = Tan (y2/x2)
  Hence argument of z1 = argument of z2