Example 1. Given $(a+1)(b+1)=2$, then the value of $\operatorname{arctg} a+\operatorname{arctg} b$ is
(A) $\frac{\pi}{3}$;
(B) $\frac{\pi}{4}$;
(C) $\frac{\pi}{6}$,
(D) $\frac{\pi}{4}$ or $-3 \pi$.
Take $a=0, b=1$, then we have $\operatorname{arctg} a+\operatorname{arctg} b$ $=\frac{\pi}{4}$,
(B) and
(D) both fit. Then take $a=-3$,
$b=-2$, then we have arctg $x+\operatorname{arctg} \dot{b}=-\frac{3 \pi}{4}$,
only (D) fits, so the answer is (D).