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Getting simple answers from Reduce, ComplexExpand and FullSimplify

Below the expression ex has roots, rts, where n in an integer and the
constant a is in the interval 0 < a <1. We may check this using FullSimplify
as follows.

ex=2 Cos[z]+I a Sin[z];

rts=z ->Pi(2n-1)/2 + I Log[(2+a)/(2-a)]/2;

FullSimplify[0 == ex/.rts,Element[n,Integers]]

My problem is to try and deduce the roots using computer algebra rather than
by hand. (I also have other expressions I would like to work on.) My attempt
at using Reduce is partially successful but leads to unfamiliar, and
difficult to interpret, ArcTanh functions

Reduce[{ex == 0,0<a<1},z]

An attempt with FullSimplify and ComplexExpand to crack the ArcTanh function
is again partially successful in giving the imaginary part I require.
However, I am stuck with Arg functions with complex arguments. The Arg
Functions are just equal to -Pi/2 but I cannot crack them further. Any
suggestions for simplifying the output from Reduce so that I get the simple
form I guessed at the start? Alternativly, are there more methods for
simplifing the complex output?


FullSimplify[ComplexExpand[Arg[2-a-I Sqrt[4-a^2]]-Arg[

2+a+I Sqrt[4-a^2]]],{0<a<1}]


Hugh Goyder

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