2.1.27 ** Find all functions \( f(x) \) and \( g(x) \) (where \( x \in \mathbf{R} \)) that satisfy
$$
\sin x + \cos y \equiv f(x) + f(y) + g(x) - g(y), \quad x, y \in \mathbf{R}
$$
In the identity, let $x=y$ to get $f(x)=\frac{\sin x+\cos x}{2}$. Thus,
$$
\sin x+\cos y \equiv \frac{\sin x+\cos x}{2}+\frac{\sin y+\cos y}{2}+g(x)-g(y) \text {. }
$$

Let $y=0$ to get $g(x)=\frac{\sin x-\cos x}{2}+\frac{1}{2}+g(0)$, i.e., $g(x)=\frac{\sin x-\cos x}{2}+c$. It is not difficult to verify that $f(x)=\frac{\sin x+\cos x}{2}, g(x)=\frac{\sin x-\cos x}{2}+c$ (where $c$ is a constant) satisfy the requirements.