# Why are some equal expressions more equal than others?

n1 = 1/3 (2 + (-Sqrt[3] + 2 Sin[(2 π)/9])/(Sqrt[3] Cos[π/9] - Sin[π/9]));
n2 = 1/2 + Sin[π/18]/(3 Cos[(2 π)/9] + Sqrt[3] Sin[(2 π)/9]);
n3 = 2/3 (1 - Cos[4 π/9]);


These three expressions are numerically equivalent. How can I coax Mathematica to simplify the first two into the third? FullSimplify only returns the original forms.

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It's at least worth pointing out that the form produced by RootReduce is the same for all three: RootReduce /@ {n1, n2, n3} – Mr.Wizard Dec 22 '12 at 5:02
Is that a political question ;-) ? – Yves Klett Dec 22 '12 at 7:18
@Yves The title, while amusing, doesn't describe the problem. Perhaps we should change it. – Mr.Wizard Dec 22 '12 at 8:19

Note that MMA aggressively evaluates n3 to 2/3 (1 - Sin[\[Pi]/18]), so I assume this is n3.

This is the best I could do:

ExpToTrig[ToRadicals[Root[MinimalPolynomial[n1, x], 2]] /. {
1 + I Sqrt[3] :> 2 (-1)^(1/3), 1 - I Sqrt[3] -> -2 (-1)^(2/3)}]

(* 2/3 - 2/3 Sin[π/18] *)


and

ExpToTrig[ToRadicals[Root[MinimalPolynomial[n2, x], 2]] /. {
1 + I Sqrt[3] :> 2 (-1)^(1/3), 1 - I Sqrt[3] -> -2 (-1)^(2/3)}]

(* 2/3 - 2/3 Sin[π/18] *)

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Excellent work around, thank you. – Phillip Dukes Dec 22 '12 at 5:37
@Phillip why note vote for his answer in that case? – Mr.Wizard Dec 22 '12 at 11:04
Ok, how do I vote for an answer? – Phillip Dukes Dec 22 '12 at 16:23