Discrete cosine transform from an even-reflected discrete Fourier transform

Reflect $x$ evenly to length $2N$ by $y_n=x_n$ and $y_{2N-1-n}=x_n$. If $Y$ is the discrete Fourier transform of $y$, thenThis reduction computes the cosine transform with one fast Fourier transform and $O(N)$ additional work.