# Simplifying this ArcTan sum expression

$Assumptions = Element[t, Reals] && Element[ω, Reals] && ω > 0 && t > 0; ArcTan[8.54596*10^7 (1. + 1. t^2 ω^2) (1.2214 + E^ω Cos[ 2 ArcTan[t ω]]), -1.70919*10^8 E^ω t ω] + ArcTan[8.54596*10^7 (1. + 1. t^2 ω^2) (1.2214 + E^ω Cos[2ArcTan[t ω]]),1.70919*10^8 E^ω t ω]//FullSimplify  Mathematica doesn't simplify this seemingly ArcTan[x,y]+ArcTan[x,-y] expression. I plotted the 3D-graph, it shows the expression is zero for all values of t and \[Omega]. I understand one underlying reason might be Mathematica isn't sure about the positivity of Cos term in x, and hence, x. Am I right? Any ideas to simplify this term in code, instead of manually, would be very much appreciated. I'm relatively new to the language. • Maybe Simplify[expr, Cos[2 ArcTan[t \[Omega]]] > 0], where expr is your expression involving Arctan? You can use the second argument of Simplify to submit assumptions. Commented Apr 30, 2018 at 10:16 • The Cos term does not necessarily have to be greater than zero. And, I have tried out inserting the global assumptions inside Simplify and FullSimply but it still doesn't work. Commented Apr 30, 2018 at 11:32 • If (1.2214 + E^\[Omega] Cos[2 ArcTan[t \[Omega]]]) is not always greater than zero, then it's not necessarily true that the ArcTan terms sum to zero. (Consider x < 0, y == 0.) Commented Apr 30, 2018 at 11:43 ## 2 Answers First you should avoid using inexact numbers if possible. expr = ArcTan[ α (1 + t^2 ω^2) (γ +E^ω Cos[2 ArcTan[t ω]]), -β E^ω t ω] + ArcTan[α(1 + t^2 ω^2) (γ + E^ω Cos[2 ArcTan[t ω]]), β E^ω t ω]  Now, with assumptions inside , the simplification evaluates as you expect Simplify[expr, Assumptions -> {ω > 0,t > 0 , α > 0, β > 0, γ >0,Cos[2 ArcTan[t ω]] > 0,Elements[{ω, t, α, β,γ}, Reals]}] (* 0 *)  • Your solution does simplify it to 0, because you're assuming Cos[2 ArcTan[t ω]] > 0. I don't see any good reason to assume so. If you do, please let me know. Commented Apr 30, 2018 at 11:46 • @ naeema18: In the question I read ... about the positivity of Cos term..  as an assumption. Commented Apr 30, 2018 at 11:53 $Assumptions = ω > 0 && t > 0;


Note that if a variable is positive it is redundant to add that it is real.

If you want exact results you must use exact numbers. Use Rationalize

expr = ArcTan[
8.54596*10^7 (1. + 1. t^2 ω^2) (1.2214 +
E^ω Cos[
2 ArcTan[t ω]]), -1.70919*10^8 E^ω t ω] +
ArcTan[8.54596*10^7 (1. + 1. t^2 ω^2) (1.2214 +
E^ω Cos[2 ArcTan[t ω]]),
1.70919*10^8 E^ω t ω] // Rationalize;


TrigToExp is what you need

expr // TrigToExp // Simplify

(* 0 *)

• If you plot Plot3D[TrigToExp[ArcTan[x, y] + ArcTan[x, -y]], {x, -1, 1}, {y, -1, 1}] you can see that the comment from Michael E2 concerning the first argument x stll holds. Commented May 1, 2018 at 9:40