$6 \cdot 148$ Find all functions $f(x)$ and $g(x) (x \in R)$ that satisfy
$$\sin x+\cos y \equiv f(x)+f(y)+g(x)-g(y), x, y \in R$$
[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 $\quad h(x)=g(x)-\frac{1}{2} \sin x+\frac{1}{2} \cos x$.
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.