In the paper it has been shown that when data vector size is odd, then the discrete cosine transform (DCT) can be computed by a real-valued DFT algorithm for appropriately permuted data samples. The same is true for the DST. Moreover, for composite transform sizes prime factor DCT algorithms can be constructed consisting of odd size real-valued DFT algorithms and DCT/DST algorithms which sizes are equal to a power of number 2. Similarly as for the DFT, the algorithms can be nested. It has been proven that the new DCT/DST algorithms require the smallest known numbers of arithmetical operations.