= Solution
Pair the two entries of the even-reflected sequence that contain $x_n$. For every $k$,
$$
\begin{aligned}
Y_k
&=\sum_{n=0}^{N-1}x_n
\left(e^{-\pi ink/N}+e^{-\pi i(2N-1-n)k/N}\right)\\
&=\sum_{n=0}^{N-1}x_n
\left(e^{-\pi ink/N}+e^{\pi i(n+1)k/N}\right)\\
&=2e^{\pi ik/(2N)}
\sum_{n=0}^{N-1}x_n
\cos\left(\frac{\pi}{N}\left(n+\frac12\right)k\right).
\end{aligned}
$$
Consequently
$$
\boxed{\frac12e^{-\pi ik/(2N)}Y_k
=\sum_{n=0}^{N-1}x_n
\cos\left(\frac{\pi}{N}\left(n+\frac12\right)k\right)}
$$
for $0\leq k<2N$, as required.
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