If the angles A, B, C of the triangle ABC be in A.P., then `(a+c)/(sqrt((a^2-ac+c^2)))=`

Options

A. `2 cos"(A+C)/2`
B. `2 sin"(A+C)/2`
C. `2 cos"(A-C)/2`
D. `2 sin"(A-C)/2`

Answer

To solve the problem, we will use the properties of triangles and the relationships between the angles. Given that the angles A, B, and C of triangle ABC are in Arithmetic Progression (A.P.), we can derive the required expression step by step.

### Step-by-Step Solution:

1. **Understanding the Angles in A.P.**:
Since A, B, and C are in A.P., we can express them as:
\[ A = B - d, \quad B = B, \quad C = B + d \]
where \(d\) is the common difference.

2. **Using the Sum of Angles in a Triangle**:
The sum of angles in a triangle is always 180 degrees:
\[ A + B + C = 180^\circ \]
Substituting the values from step 1:
\[ (B - d) + B + (B + d) = 180^\circ \]
This simplifies to:
\[ 3B = 180^\circ \implies B = 60^\circ \]

3. **Finding A and C**:
Now that we have \(B\):
\[ A = 60^\circ - d, \quad C = 60^\circ + d \]
The angles can be expressed as:
\[ A + C = (60^\circ - d) + (60^\circ + d) = 120^\circ \]

4. **Using the Law of Sines**:
By the Law of Sines, we have:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k \]
Thus, we can express \(a\) and \(c\) in terms of \(k\):
\[ a = k \sin A, \quad c = k \sin C \]

5. **Finding the Expression**:
We need to evaluate:
\[ \frac{a + c}{\sqrt{a^2 - ac + c^2}} \]
Substituting the values of \(a\) and \(c\):
\[ a + c = k \sin A + k \sin C = k (\sin A + \sin C) \]

6. **Calculating \(a^2 - ac + c^2\)**:
We know:
\[ a^2 = (k \sin A)^2, \quad c^2 = (k \sin C)^2, \quad ac = k^2 \sin A \sin C \]
Therefore:
\[ a^2 + c^2 = k^2 (\sin^2 A + \sin^2 C) \]
Now substituting into the expression:
\[ a^2 - ac + c^2 = k^2 (\sin^2 A + \sin^2 C - \sin A \sin C) \]

7. **Final Expression**:
Thus, we can write:
\[ \frac{a + c}{\sqrt{a^2 - ac + c^2}} = \frac{k (\sin A + \sin C)}{\sqrt{k^2 (\sin^2 A + \sin^2 C - \sin A \sin C)}} \]
This simplifies to:
\[ \frac{\sin A + \sin C}{\sqrt{\sin^2 A + \sin^2 C - \sin A \sin C}} \]

### Final Result:
The final expression simplifies to:
\[ \frac{a+c}{\sqrt{a^2 - ac + c^2}} = 2 \]