Why are some equal expressions more equal than others?

Question

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.

Answer 1

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] *)