6. (2002 National Mathematics Competition Shandong Province Preliminary Question) Given the function $f(x)=\sin (\omega x+\varphi)(\omega>0, x \in \mathbf{R})$, which satisfies $f(x)=f(x+1)-f(x+2)$, if $A=\sin (\omega x+\varphi+9 \omega) \cdot B=\sin (\omega x+\varphi-9 \omega)$, then the relationship between $A$ and $B$ is
A. $A>B$
B. $A=B$
C. $A<B$
D. Uncertain
6. B This problem clearly requires deriving the function's period using the known equations to reach the correct conclusion.

From $f(x)=f(x+1)-f(x+2), f(x+1)=f(x+2)-f(x+3)$, adding these equations gives $f(x+3)=-f(x), x \in \mathbf{R}$, then $f(x+6)=-f(x+3)=f(x), x \in \mathbf{R}$, thus $f(x)$ is a periodic function with a period of 6.
$$
A=f(x+9)=f(x+3)=f(x-3)=f(x-9)=B \text {, choose B. }
$$