Multivariate trigonometric series are used for analytical and numerical solutions of a wide class of applied problems. This mathematical tool is in demand in celestial mechanics, quantum mechanics, electrical engineering, acoustics, optics, and the theory of signal and image processing. Multiplying such series when implementing numerical methods is a complex task that requires a large amount of machine time and memory. The formulas for the coefficients of the product of multivariate series derived in this article, can significantly reduce memory consumption and calculation time. Therefore, the use of these formulas in the software development to solve the mentioned tasks can improve its quality and competitiveness. All the results obtained are presented in the conclusion as final formulas for software implementation.