How to simplify this complex equation

Questions
1

IMG_1021.jpg
IMG_1021.jpg1308×424 85.3 KB

2

I forgot to add, I need to simplify the top equation so I am left with the bottom equation.

3

Exp[I v] == SolveValues[(z - 1)/(z + 1) == (1 + x)/(1 - x)*(Exp[I E] - 1)/(Exp[I E] + 1), z][[1]]

(*Where:   z=Exp[I v]*)

(*E^(I v) == -((E^(I E) - x)/(-1 + E^(I E) x))*)

Another way:

 func = (Exp[I v] - 1)/(Exp[I v] + 1) - (1 + x)/(1 - x)*(Exp[I E] - 1)/(Exp[I E] + 1) // FullSimplify
 Sol = Solve[func == 0, v]
 (v /. Sol /. C[1] -> 0)[[1]]
 Exp[I*v] == Exp[I*((v /. Sol /. C[1] -> 0)[[1]])]

 (*E^(I v) == -((E^(I E) - x)/(-1 + E^(I E) x))*)

Regards M.I.

4

Here is a step-by-step derivation:

(E^(I v) - 1)/(E^(I v) + 1) ==
 (1 + x)/(1 - x)*(Exp[I E] - 1)/(Exp[I E] + 1)
MultiplySides[%, E^(I v) + 1, GenerateConditions -> False]
Collect[%, E^(I v)]
AddSides[%, 1 - %[[2, 2]]]
Collect[%, E^(I v), Factor]
DivideSides[%, %[[1]]/E^(I v), GenerateConditions -> False]