In a `DeltaABC,` the tangent of hald the difference of two angle is one-third the tangent of half the sum of the angle. Determine the ratio of the sides opposite to the angles.

Answer

To solve the problem, we need to determine the ratio of the sides opposite to the angles in triangle \( ABC \) given that the tangent of half the difference of two angles is one-third the tangent of half the sum of the angles. Let's denote the angles as \( A \), \( B \), and \( C \).

### Step-by-Step Solution:

1. **Understanding the Given Condition:**
We are given that:
\[ \tan\left(\frac{A - B}{2}\right) = \frac{1}{3} \tan\left(\frac{A + B}{2}\right) \]

2. **Using the Sum of Angles in a Triangle:**
Since the sum of angles in a triangle is \( \pi \):
\[ A + B + C = \pi \implies A + B = \pi - C \]

3. **Substituting into the Equation:**
Substitute \( A + B = \pi - C \) into the equation:
\[ \tan\left(\frac{A - B}{2}\right) = \frac{1}{3} \tan\left(\frac{\pi - C}{2}\right) \]
We know that \( \tan\left(\frac{\pi - C}{2}\right) = \cot\left(\frac{C}{2}\right) \), so:
\[ \tan\left(\frac{A - B}{2}\right) = \frac{1}{3} \cot\left(\frac{C}{2}\right) \]

4. **Using the Tangent Half-Angle Formula:**
By the tangent half-angle formula, we can express:
\[ \tan\left(\frac{A - B}{2}\right) = \frac{A - B}{A + B} \]
Thus, we have:
\[ \frac{A - B}{A + B} = \frac{1}{3} \cot\left(\frac{C}{2}\right) \]

5. **Expressing \( \cot\left(\frac{C}{2}\right) \):**
Recall that:
\[ \cot\left(\frac{C}{2}\right) = \frac{1}{\tan\left(\frac{C}{2}\right)} \]
Therefore, substituting this back gives us:
\[ \frac{A - B}{A + B} = \frac{1}{3} \cdot \frac{1}{\tan\left(\frac{C}{2}\right)} \]

6. **Cross-Multiplying:**
Cross-multiplying gives:
\[ 3(A - B) = (A + B) \cdot \cot\left(\frac{C}{2}\right) \]

7. **Rearranging the Equation:**
Rearranging leads to:
\[ 3A - 3B = A \cdot \cot\left(\frac{C}{2}\right) + B \cdot \cot\left(\frac{C}{2}\right) \]

8. **Setting Up Ratios:**
From the equation \( 3A - 3B = A \cdot \cot\left(\frac{C}{2}\right) + B \cdot \cot\left(\frac{C}{2}\right) \), we can isolate terms to find:
\[ 3A - 4B = 0 \implies 3A = 4B \implies \frac{A}{B} = \frac{4}{3} \]

9. **Final Ratio of Sides:**
The sides opposite to angles \( A \) and \( B \) are in the ratio:
\[ \frac{a}{b} = \frac{4}{3} \]

### Conclusion:
The ratio of the sides opposite to angles \( A \) and \( B \) is \( 4:3 \).