1-z = 1 - cosx - isinx

1+z = 1+cosx+isinx

Times top and bottom by 1+cosx-isinx

Numerator = (1-cosx-isinx)(1+cosx-isinx)

= 1+cosx-isinx-cosx-cos^2x+isinxcosx-isinx-isinxcosx-sin^2x

= 1-2isinx-(sin^2x+cos^2x)

= -2isinx

Denominator = (1+cosx+isinx)(1+cosx-isinx)

= (1+cosx)^2 - (isinx)^2

= 1+cos^2x+2cosx+sin^2x

= 2(1+cosx)

(1-z)/(1+z) = -isinx/(1+cosx)

= -i(2sin(x/2)cos(x/2))/(1+1-2sin^2(x/2))

= -i(sin(x/2)cos(x/2))/(cos^2(x/2))

= -isin(x/2)/cos(x/2)

= -itan(x/2)

In summary: times top and bottom by conjugate of the denominator, i.e. realising it.

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