If in a `DeltaABC,`
`sin A: sin C = sin (A - B): sin (B-C),` then `a^(2), b^(2), c^(2)`are in

Options

A. A.P
B. G.P.
C. H.P.
D. none of these

Explanation

To solve the problem, we start with the given relationship in triangle \( \Delta ABC \):

\[ \frac{\sin A}{\sin C} = \frac{\sin(A - B)}{\sin(B - C) \]

### Step 1: Rewrite the Sine Functions
Using the sine addition and subtraction formulas, we can express \( \sin A \) and \( \sin C \) in terms of the other angles:
- \( \sin A = \sin(\pi - (B + C)) = \sin(B + C) \)
- \( \sin C = \sin(A + B) \)

### Step 2: Substitute into the Equation
Substituting these into the original equation gives:

\[ \frac{\sin(B + C)}{\sin(A + B)} = \frac{\sin(A - B)}{\sin(B - C)} \]

### Step 3: Cross-Multiply
Cross-multiplying the equation results in:

\[ \sin(B + C) \cdot \sin(B - C) = \sin(A + B) \cdot \sin(A - B) \]

### Step 4: Use the Sine Difference Identity
Using the identity \( \sin x \cdot \sin y = \frac{1}{2} [\cos(x - y) - \cos(x + y)] \), we can express both sides:

- Left Side:
\[ \sin(B + C) \cdot \sin(B - C) = \frac{1}{2} [\cos(2B) - \cos(2C)] \]

- Right Side:
\[ \sin(A + B) \cdot \sin(A - B) = \frac{1}{2} [\cos(2A) - \cos(2B)] \]

### Step 5: Set the Equations Equal
Setting the two sides equal gives:

\[ \frac{1}{2} [\cos(2B) - \cos(2C)] = \frac{1}{2} [\cos(2A) - \cos(2B)] \]

### Step 6: Simplify
Multiplying through by 2 and rearranging leads to:

\[ \cos(2B) + \cos(2B) = \cos(2A) + \cos(2C) \]

This simplifies to:

\[ 2\cos(2B) = \cos(2A) + \cos(2C) \]

### Step 7: Apply the Sine Rule
Using the sine rule \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k \), we can express \( a, b, c \) in terms of \( k \):

\[ a = k \sin A, \quad b = k \sin B, \quad c = k \sin C \]

### Step 8: Square the Sides
Squaring these gives:

\[ a^2 = k^2 \sin^2 A, \quad b^2 = k^2 \sin^2 B, \quad c^2 = k^2 \sin^2 C \]

### Step 9: Establish the Relationship
From the earlier derived relationship, we can conclude that:

\[ a^2 + c^2 = 2b^2 \]

This indicates that \( a^2, b^2, c^2 \) are in Arithmetic Progression (AP).

### Final Result
Thus, the final result is:

\[ a^2, b^2, c^2 \text{ are in AP.} \]

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