Of sum of cosines and the $7$th roots of unity In my solution here, it was shown that 
$$\omega+\omega^2+\omega^4=-\frac 12\pm\frac{\sqrt7}2\qquad\qquad (\omega=e^{i2\pi/7})$$
from which we know that 
$$\sin \frac{2\pi}7+\sin \frac{4\pi}7-\sin \frac{6\pi}7=\Im (\omega+\omega^2+\omega^4)=\frac{\sqrt7}2$$
This can also be verified easily by computation. 
By the same token it would appear that 
$$\cos \frac{2\pi}7+\cos \frac{4\pi}7-\cos \frac{6\pi}7=\Re (\omega+\omega^2+\omega^4)=\frac 12$$
However a quick computational check shows that the result is $1.3019...$ and not $\frac 12$.
Why is this so?

You're starting from the wrong assumption that $-\omega^3=\omega^4$, which is obviously wrong, because it would imply $\omega=-1$.
The true fact is that
$$
-\sin\frac{6\pi}{7}=\sin\frac{8\pi}{7}
$$
but
$$
\cos\frac{6\pi}{7}=\cos\frac{8\pi}{7}
$$
because
$$
\frac{6\pi}{7}+\frac{8\pi}{7}=2\pi
$$
and
$$
\sin(2\pi-\alpha)=-\sin\alpha,\qquad
\cos(2\pi-\alpha)=\cos\alpha
$$
