Find the value of $$E=\sum_{k=1}^4 \big(\cos\frac{k \pi}{9}\big)^4.$$
Source: list of problems related to the preparation for math contests. No answer provided.
My attempt: I looked into $$\sum_{k=1}^4 z^{k4}$$ for $z=cis(\pi/9)$, but this doesn't appear to be useful.
Hints and full answers are appreciated. Sorry if this is a duplicate.