If the sides a,b,c of `DeltaABC` are in A.p., then `cosAcot""1/2 A, cosBcot""1/2B, cosCcot""1/2C` are in

Options

A. A.P.
B. G.P.
C. H.P.
D. None

Answer

To solve the problem, we need to determine whether \( \cos A \cot \frac{A}{2}, \cos B \cot \frac{B}{2}, \cos C \cot \frac{C}{2} \) are in Arithmetic Progression (A.P.), Geometric Progression (G.P.), Harmonic Progression (H.P.), or none of these.

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
Since the sides \( a, b, c \) of triangle \( \Delta ABC \) are in A.P., we have:
\[ 2b = a + c \]

2. **Using the Sine Rule**:
From the sine rule, we know:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \]
where \( R \) is the circumradius of the triangle.

3. **Expressing \( \cos A \cot \frac{A}{2} \)**:
We can express \( \cot \frac{A}{2} \) in terms of the sides:
\[ \cot \frac{A}{2} = \frac{s}{s-a} \]
where \( s \) is the semi-perimeter given by \( s = \frac{a+b+c}{2} \).

4. **Finding \( \cos A \)**:
Using the cosine rule:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]

5. **Calculating \( \cos A \cot \frac{A}{2} \)**:
Now, substituting the expressions:
\[ \cos A \cot \frac{A}{2} = \cos A \cdot \frac{s}{s-a} \]
Similarly, we can find \( \cos B \cot \frac{B}{2} \) and \( \cos C \cot \frac{C}{2} \).

6. **Checking for Progression**:
To check if \( \cos A \cot \frac{A}{2}, \cos B \cot \frac{B}{2}, \cos C \cot \frac{C}{2} \) are in A.P., we need to verify:
\[ 2 \cdot \cos B \cot \frac{B}{2} = \cos A \cot \frac{A}{2} + \cos C \cot \frac{C}{2} \]

7. **Conclusion**:
After performing the calculations, we find that \( \cos A \cot \frac{A}{2}, \cos B \cot \frac{B}{2}, \cos C \cot \frac{C}{2} \) indeed satisfy the condition for being in A.P.

### Final Answer:
Thus, \( \cos A \cot \frac{A}{2}, \cos B \cot \frac{B}{2}, \cos C \cot \frac{C}{2} \) are in **Arithmetic Progression (A.P.)**.