To solve the problem, we need to establish the relationship between \(x\), \(y\), and \(z\) when they are in Arithmetic Progression (A.P.) and also when \(\tan^{-1} x\), \(\tan^{-1} y\), and \(\tan^{-1} z\) are in A.P.
### Step-by-Step Solution:
1. **Understanding A.P. Condition**:
Since \(x\), \(y\), and \(z\) are in A.P., we can write:
\[
2y = x + z \quad \text{(Equation 1)}
\]
**Hint**: Recall that for three numbers to be in A.P., the middle number must be the average of the other two.
2. **Applying A.P. to Inverse Tangents**:
Since \(\tan^{-1} x\), \(\tan^{-1} y\), and \(\tan^{-1} z\) are also in A.P., we have:
\[
2\tan^{-1} y = \tan^{-1} x + \tan^{-1} z
\]
**Hint**: Use the property of A.P. for the angles, which states that the middle angle is the average of the other two.
3. **Using the Formula for Tangent Addition**:
We can use the formula for the tangent of the sum of two angles:
\[
\tan(\tan^{-1} x + \tan^{-1} z) = \frac{x + z}{1 - xz}
\]
Thus, we can rewrite the equation:
\[
\tan(2\tan^{-1} y) = \frac{2y}{1 - y^2}
\]
Therefore, we equate:
\[
\frac{2y}{1 - y^2} = \frac{x + z}{1 - xz}
\]
**Hint**: Remember the double angle formula for tangent, which is useful when dealing with \(\tan^{-1}\).
4. **Substituting Equation 1**:
From Equation 1, we know \(x + z = 2y\). Substituting this into our equation gives:
\[
\frac{2y}{1 - y^2} = \frac{2y}{1 - xz}
\]
**Hint**: Simplifying the equation can help isolate terms.
5. **Cross Multiplying**:
Cross multiplying gives:
\[
2y(1 - xz) = 2y(1 - y^2)
\]
Assuming \(y \neq 0\), we can divide both sides by \(2y\):
\[
1 - xz = 1 - y^2
\]
**Hint**: Be careful with division by zero; ensure \(y\) is not zero.
6. **Simplifying the Equation**:
This simplifies to:
\[
y^2 = xz
\]
**Hint**: This is a key result that connects the A.P. and G.P. conditions.
7. **Conclusion**:
The condition \(y^2 = xz\) indicates that \(x\), \(y\), and \(z\) are in Geometric Progression (G.P.) as well. Therefore, if \(x\), \(y\), and \(z\) are in A.P. and \(\tan^{-1} x\), \(\tan^{-1} y\), and \(\tan^{-1} z\) are also in A.P., it follows that \(x\), \(y\), and \(z\) must be in G.P.
**Final Answer**: The only possible solution is that \(x\), \(y\), and \(z\) are in G.P.
