In `DeltaABC` if `(sinA)/(sinC)=(sin(A-B))/(sin(B-C))`, then `a^(2), b^(2), c^(2)` are in :

Options

A. a,b,c are in A.P.
B. `a^(2), b^(2),c^(2)` are in A.P.
C. a,b,c are in H.P.
D. `a^(2), b^(2), c^(2)` are in H.P.

Answer

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

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

### Step 1: Rewrite Angles
Using the property of angles in a triangle, we know that \( A + B + C = \pi \). Therefore, we can express \( A \) and \( C \) in terms of \( B \):

\[ A = \pi - (B + C) \quad \text{and} \quad C = \pi - (A + B) \]

### Step 2: Substitute into Sine Functions
Now, we can substitute these into the sine functions:

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

### Step 3: Substitute into the Given Equation
Substituting these into the original equation gives us:

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

### Step 4: Cross Multiply
Cross-multiplying yields:

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

### Step 5: Use Sine Addition and Subtraction Formulas
Using the sine addition and subtraction formulas, we can expand both sides:

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

### Step 6: Set Up the Equation
Substituting these expansions into our equation gives:

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

### Step 7: Simplify Using Product-to-Sum Identities
Using product-to-sum identities, we can simplify both sides. After simplification, we will find a relation involving \( \sin^2 A, \sin^2 B, \sin^2 C \).

### Step 8: Establish the Relationship
From the derived equation, we can conclude that:

\[ 2 \sin^2 B = \sin^2 A + \sin^2 C \]

This implies that \( \sin^2 A, \sin^2 B, \sin^2 C \) are in arithmetic progression (AP).

### Step 9: Apply Sine Rule
Using the sine rule, we know:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

If \( \sin^2 A, \sin^2 B, \sin^2 C \) are in AP, then \( a^2, b^2, c^2 \) must also be in AP.

### Conclusion
Thus, we conclude that \( a^2, b^2, c^2 \) are in arithmetic progression (AP).

### Final Answer
The answer is that \( a^2, b^2, c^2 \) are in AP.

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