Example 8 Let $\arctan a+\arctan b+\arctan c+\frac{\pi}{2}=0$, try to compare the size of the two numbers in each of the following groups:
(1) $a b+b c+c a$ and 1;
(2) $a+b+c$ and $a b c$.

保留了原文的换行和格式，以下是翻译结果：

Example 8 Let $\arctan a+\arctan b+\arctan c+\frac{\pi}{2}=0$, try to compare the size of the two numbers in each of the following groups:
(1) $a b+b c+c a$ and 1;
(2) $a+b+c$ and $a b c$.
Solve: Regarding $\arctan a, \arctan b$ and $\arctan c$ as the principal values of the arguments of the complex numbers $z_{1}=1+a \mathrm{i}, z_{2}=1+b \mathrm{i}$ and $z_{3}=$ $1+c \mathrm{i}$, let $z=z_{1} z_{2} z_{3}, z=(1+a \mathrm{i})(1+b \mathrm{i})(1+c \mathrm{i})=(1-a b-b c-c a)$ $+(a+b+c-a b c) \mathrm{i}$.

According to the geometric meaning of complex number multiplication, the original expression is transformed into $\arg z=-\frac{\pi}{2}$, from which it is known that the real part of the complex number $z$ is zero, and the imaginary part is negative.
Thus, we have (1) $a b+b c+c a=1$; (2) $a+b+c<a b c$.