6. 148 Find all functions \( f(x) \) and \( g(x) \) that satisfy
\[
\sin x + \cos y \equiv f(x) + f(y) + g(x) - g(y), \quad x, y \in \mathbb{R}.
\]
[Solution] In the identity, let $x=y$ to get
$$
f(x)=\frac{\sin x+\cos x}{2} .
$$

Thus, $\sin x+\cos y \equiv \frac{\sin x+\cos x}{2}+\frac{\sin y+\cos y}{2}+g(x)-g(y)$
which means $g(x)-\frac{1}{2} \sin x+\frac{1}{2} \cos x \equiv g(y)-\frac{1}{2} \sin y+\frac{1}{2} \cos y$.
Let $h(x)=g(x)-\frac{1}{2} \sin x+\frac{1}{2} \cos x$.
Then for any $x, y \in \mathbb{R}$, we have $h(x)=h(y)$. This indicates that $h(x) \equiv C$. Therefore, the function pairs that satisfy the conditions in the problem are
$$
f(x)=\frac{\sin x+\cos x}{2}, g(x)=\frac{\sin x-\cos x}{2}+c .
$$

where $c$ is a constant.