We present algorithms for the discrete cosine transform (DCT) and discrete sine transform (DST), of types II and III, that achieve a lower count of real multiplications and additions than previously published algorithms, without sacrificing numerical accuracy. Asymptotically, the operation count is reduced from ~ 2N log_2 N to ~ (17/9) N log_2 N for a power-of-two transform size N. Furthermore, we show that a further N multiplications may be saved by a certain rescaling of the inputs or outputs, generalizing a well-known technique for N=8 by Arai et al. These results are derived by considering the DCT to be a special case of a DFT of length 4N, with certain symmetries, and then pruning redundant operations from a recent improved fast Fourier transform algorithm (based on a recursive rescaling of the conjugate-pair split radix algorithm). The improved algorithms for DCT-III, DST-II, and DST-III follow immediately from the improved count for the DCT-II.