In a `Delta ` ABC the tangent of half the difference of two angles is one thired the tangement of half the sum of the angles .Determine the ratio of the sides opposite to the angles .

Explanation

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.

### Step-by-Step Solution:

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