How to evaluate $\left(\cos{\frac{5\pi}{9}}\right)^{11}+\left(\cos{\frac{7\pi}{9}}\right)^{11}+\left(\cos{\frac{11\pi}{9}}\right)^{11}$ How to evaluate $$\left(\cos{\frac{5\pi}{9}}\right)^{11}+\left(\cos{\frac{7\pi}{9}}\right)^{11}+\left(\cos{\frac{11\pi}{9}}\right)^{11}?$$
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 A: Starting from Frieder's final expression, there is a terrible closed form since $$\cos \left(\frac{\pi }{9}\right)=\frac{\sqrt[3]{1-i \sqrt{3}}+\sqrt[3]{1+i \sqrt{3}}}{2 \sqrt[3]{2}}$$ $$\cos \left(\frac{2\pi }{9}\right)=2\cos^2 \left(\frac{\pi }{9}\right)-1$$ $$\sin \left(\frac{\pi }{18}\right)=\sqrt{\frac{1-\cos \left(\frac{\pi }{9}\right)}{2}}$$ After Lucian's comment, this is $$\frac{231}{512}-\frac{\sqrt[3]{\frac{3}{2}} \left(1-i \sqrt{3}\right)
   \sqrt[3]{A}}{4096}-\frac{132463\ 3^{2/3} \left(1+i \sqrt{3}\right)}{2048\ 2^{2/3}
   \sqrt[3]{A}}$$ with $A=-42951219+93177701 i \sqrt{3}$.
A: Using for an odd $n$
$${\cos ^n}(x) = \frac{1}{{{2^{n - 1}}}}\sum\limits_{k = 0}^{\frac{{n - 1}}{2}} {\left( \begin{gathered}
  n \\ 
  k \\ 
\end{gathered}  \right)} \cos ((n - 2k)x)$$
gives me:
$${\cos ^{11}}(t) = \frac{{462\cos (t) + 330\cos (3t) + 165\cos (5t) + 55\cos (7t) + 11\cos (9t) + \cos (11t)}}{{1024}}$$
It follows therfore:
$${\cos ^{11}}\left( {\frac{{5\pi }}{9}} \right) = \frac{{154 - 462\sin \left( {\frac{\pi }{{18}}} \right) + 56\cos \left( {\frac{\pi }{9}} \right) - 165\cos \left( {\frac{{2\pi }}{9}} \right)}}{{1024}}$$
$${\cos ^{11}}\left( {\frac{{7\pi }}{9}} \right) = \frac{{154 - 56\sin \left( {\frac{\pi }{{18}}} \right) + 165\cos \left( {\frac{\pi }{9}} \right) - 462\cos \left( {\frac{{2\pi }}{9}} \right)}}{{1024}}$$
and
$${\cos ^{11}}\left( {\frac{{11\pi }}{9}} \right) = \frac{{154 - 56\sin \left( {\frac{\pi }{{18}}} \right) + 165\cos \left( {\frac{\pi }{9}} \right) - 462\cos \left( {\frac{{2\pi }}{9}} \right)}}{{1024}}$$
summing up:
$${\cos ^{11}}\left( {\frac{{5\pi }}{9}} \right) + {\cos ^{11}}\left( {\frac{{7\pi }}{9}} \right) + {\cos ^{11}}\left( {\frac{{11\pi }}{9}} \right) = \frac{{462 - 574\sin \left( {\frac{\pi }{{18}}} \right) + 386\cos \left( {\frac{\pi }{9}} \right) - 1089\cos \left( {\frac{{2\pi }}{9}} \right)}}{{1024}}$$
$$ \approx  - 0.10661630948635201594863746012996982756343832484191$$
