Show that the discrete cosine transform of a length- \(N\) sequence \(x(n)\) can be computed from the length \(N\) DFT of a sequence \(\hat{x}(n)\) consisting of the following reordering of the even and odd elements of \(x(n)\) :

\[\left.\begin{array}{rl}\hat{x}(n) & =x(2 n) \\\hat{x}(N-1-n) & =x(2 n+1)\end{array}\right\} \text { for } 0 \leq n \leq \frac{N}{2}-1\]