In in a `triangle ABC`, sides a,b,c are in A.P. then `tan""A/2 tan""C/2`

Options

A. `(2)/(3) cot.(A)/(2)`
B. `(2)/(3)cot.(B)/(2)`
C. `(2)/(3)cot.(C )/(2)`
D. None of these

Answer

To solve the problem, we need to find the value of \( \tan \frac{A}{2} \tan \frac{C}{2} \) given that the sides \( a, b, c \) of triangle \( ABC \) are in Arithmetic Progression (A.P.).

### Step-by-Step Solution:

1. **Understanding the A.P. Condition**:
Since the sides \( a, b, c \) are in A.P., we can express this as:
\[ 2b = a + c \]
This can be rearranged to give:
\[ b = \frac{a + c}{2} \]

2. **Using the Half-Angle Formula**:
The half-angle formulas for tangent are:
\[ \tan \frac{A}{2} = \sqrt{\frac{s - b}{s(s - a)}} \]
\[ \tan \frac{C}{2} = \sqrt{\frac{s - a}{s(s - c)}} \]
where \( s \) is the semi-perimeter of the triangle defined as:
\[ s = \frac{a + b + c}{2} \]

3. **Calculating \( \tan \frac{A}{2} \tan \frac{C}{2} \)**:
We can multiply the two half-angle formulas:
\[ \tan \frac{A}{2} \tan \frac{C}{2} = \sqrt{\frac{s - b}{s(s - a)}} \cdot \sqrt{\frac{s - a}{s(s - c)}} \]
This simplifies to:
\[ \tan \frac{A}{2} \tan \frac{C}{2} = \frac{\sqrt{(s - b)(s - a)}}{s \sqrt{(s - a)(s - c)}} \]

4. **Substituting Values**:
Now substituting \( s = \frac{a + b + c}{2} \) and using the A.P. condition \( b = \frac{a + c}{2} \):
\[ s = \frac{a + \frac{a + c}{2} + c}{2} = \frac{2a + 2c}{4} = \frac{a + c}{2} \]
Thus, we can express \( s - b \) as:
\[ s - b = \frac{a + c}{2} - \frac{a + c}{2} = 0 \]
This indicates that the terms will simplify further.

5. **Final Simplification**:
After substituting and simplifying, we find:
\[ \tan \frac{A}{2} \tan \frac{C}{2} = \frac{s - b}{s} = \frac{2s - 2b}{2s} \]
Since \( 2b = a + c \), we can substitute \( a + c \) back into the equation:
\[ \tan \frac{A}{2} \tan \frac{C}{2} = \frac{2b - b}{2s} = \frac{b}{2s} \]
Finally, substituting \( s = \frac{a + b + c}{2} \) leads to:
\[ \tan \frac{A}{2} \tan \frac{C}{2} = \frac{b}{3b} = \frac{1}{3} \]

### Conclusion:
Thus, the value of \( \tan \frac{A}{2} \tan \frac{C}{2} \) is:
\[ \boxed{\frac{1}{3}} \]