Step 1: Understanding the Concept:
We are given an equation involving the sum of three inverse tangent functions.
We can use the standard trigonometric identity for the tangent of a sum of three angles.
Step 2: Key Formula or Approach:
If $A + B + C = \frac{\pi}{2}$, then $\tan(A + B + C)$ is undefined (approaches infinity).
The formula for $\tan(A+B+C)$ is:
\[ \tan(A+B+C) = \frac{\sum \tan A - \prod \tan A}{1 - \sum \tan A \tan B} \]
For the fraction to approach infinity, the denominator must be zero:
$1 - (\tan A \tan B + \tan B \tan C + \tan C \tan A) = 0$
$\Rightarrow \tan A \tan B + \tan B \tan C + \tan C \tan A = 1$.
Step 3: Detailed Explanation:
Let $A = \tan^{-1}\left(\frac{x}{2}\right) \Rightarrow \tan A = \frac{x}{2}$
Let $B = \tan^{-1}\left(\frac{y}{2}\right) \Rightarrow \tan B = \frac{y}{2}$
Let $C = \tan^{-1}\left(\frac{z}{2}\right) \Rightarrow \tan C = \frac{z}{2}$
Given that $A + B + C = \frac{\pi}{2}$.
As derived above, this implies:
\[ \tan A \tan B + \tan B \tan C + \tan C \tan A = 1 \]
Substitute the values of $\tan A, \tan B, \tan C$:
\[ \left(\frac{x}{2}\right)\left(\frac{y}{2}\right) + \left(\frac{y}{2}\right)\left(\frac{z}{2}\right) + \left(\frac{z}{2}\right)\left(\frac{x}{2}\right) = 1 \]
\[ \frac{xy}{4} + \frac{yz}{4} + \frac{zx}{4} = 1 \]
Multiply the entire equation by 4:
\[ xy + yz + zx = 4 \]
Step 4: Final Answer:
The value of $xy + yz + zx$ is $4$.