Example 1 Find all functions $f(x)$ and $g(x)$ that satisfy
$$
f(x)+f(y)+g(x)-g(y) \geqslant \sin x+\cos y, x, y \in \mathbf{R}
$$
Solve the separation function, separating the parts containing $x, y$ on both sides of the inequality,
$$
f(x)+g(x)-\sin x \geqslant g(y)-f(y)+\cos y,
$$

Introduce auxiliary functions $p(x), q(y), x, y \in \mathbf{R}$, and let
$$
p(x)=f(x)+g(x)-\sin x, q(y)=g(y)-f(y)+\cos y .
$$

where $p(x) \geqslant q(y), \forall x, y \in \mathbf{R}$.
In equation (1), let $y=x$. Adding and subtracting the two equations, we get
$$
f(x)=\frac{1}{2}[\sin x+\cos x+p(x)-q(x)], g(x)=\frac{1}{2}[\sin x-\cos x+p(x)+q(x)] \text {, }
$$

where $p(x) \geqslant q(y), \forall x, y \in \mathbf{R}$.