What is the Discrete Cosine Transform and its applications?

DCT transforms signal to frequency domain using only cosine basis functions (real-valued, unlike complex DFT). DCT-II (most common): X[k] = sum(x[n]*cos(pi*k*(2n+1)/(2N))). Properties: Energy compaction (most energy in few low-frequency coefficients), Real coefficients (efficient for real signals), and Reduced boundary artifacts vs DFT. Applications: JPEG image compression (2D DCT of 8x8 blocks), video compression (MPEG, H.264), audio compression (MP3 uses modified DCT), and speech coding. DCT enables lossy compression by quantizing/discarding small high-frequency coefficients.