Novel Series Expansion of sin(x) by Synthesizing the Weierstrass Product with Explicit Zeta Function Components

In this paper, we derive explicit exponential series representations for the sine function involving even values of the Riemann zeta function. The result is obtained via logarithmic differentiation and integration of the Weierstrass product. We further demonstrate that the method extends naturally to the cosine function, yielding an analogous representation. These results highlight a structural connection between trigonometric entire functions and the distribution of their zeros.