Discrete Cosine Transform (DCT) and Discrete Sine Transform (DST) are widely used in image and data compression applications. To process the DCT or DST of a signal a portion of length N is extracted by windowing. By shifting the window point by point the entire signal can be processed. The algorithms are developed that are capable of updating the DCT and DST independently to reflect the modified window contents i.e. for calculating the DCT of the shifted sequence no DST coefficients are used and similarly for calculating the DST of the shifted sequence no DCT coefficients are used. These algorithms constitute an improvement over previous DCT/DST update algorithms as it establishes independence between the DCT and the DST. The update algorithms used to calculate the transform of the shifted sequence uses less computation as compared to directly evaluating the modified transform via standard fast transform algorithms. Firstly, the r-point, 1 ≤ r ≤ N-1, update algorithms are derived in the presence of the rectangular window. Thereafter, one point independent windowed update in the presence of split-triangular, Hanning, Hamming and Blackman windows are developed. The algorithms were implemented in C language to test their correctness. Thereafter the hardware circuits capable of computing the independent update of DCT-II for the rectangular window of size N=8 and step size of 1 and 4 are developed. The windowed update algorithms are derived for DCT and DST type-I through IV, however the hardware implementation of type-II is given as it is the most frequently used transform.