On the general solution and stability of the functional equation \(f (x-y) - f(x) f(y) = dsinxsiny\)

This paper is concerned with the investigation of the general solution to the functional equation \( f(x - y) - f(x)f(y) = d \sin x \sin y \quad \text{for all } x, y \in \mathbb{R}, \) where \(f: \mathbb{R} \to \mathbb{R}\) is an unknown function and \(d \in \mathbb{R} \setminus \{0\}\) is a real constant satisfying \(d < 1\). This nonlinear functional equation establishes an intriguing interplay between multiplicative and additive behaviors of the function \(f\), perturbed by a bounded trigonometric term. We provide a complete characterization of all real-valued functions satisfying the equation, under minimal regularity assumptions. In addition, we analyze the Hyers–Ulam stability and the so-called superstability of the equation in the sense of functional equations, establishing that approximate solutions under certain bounded perturbations necessarily converge to exact solutions.