In `DeltaABC` if `(sinA)/(sinC)=(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

Answer

To solve the problem, we start with the given equation in triangle ABC:

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

### Step 1: Rewrite angle A
Using the identity \(A = \pi - (B + C)\), we can rewrite \(A\) in terms of \(B\) and \(C\):

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

### Step 2: Use sine addition formula
Now, we apply the sine addition formula:

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

### Step 3: Rewrite the equation
Substituting this back into the original equation gives:

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

### Step 4: Rewrite \(A - B\) and \(B - C\)
Next, we rewrite \(A - B\) and \(B - C\):

\[ A - B = \pi - (B + C) - B = \pi - 2B - C \]
\[ B - C = B - C \]

Thus, we have:

\[ \sin(A - B) = \sin(\pi - 2B - C) = \sin(2B + C) \]

### Step 5: Use sine addition formula again
Using the sine addition formula again:

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

### Step 6: Substitute back
Substituting this into our equation gives:

\[ \frac{\sin B \cos C + \cos B \sin C}{\sin C} = \frac{\sin(2B) \cos C + \cos(2B) \sin C}{\sin(B - C)} \]

### Step 7: Cross-multiply
Cross-multiplying gives:

\[ (\sin B \cos C + \cos B \sin C) \sin(B - C) = (\sin(2B) \cos C + \cos(2B) \sin C) \sin C \]

### Step 8: Simplify and analyze
After simplifying, we can analyze the resulting equation to find relationships between \(a^2\), \(b^2\), and \(c^2\).

### Step 9: Use the Sine Rule
Using the sine rule \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \), we can express \(a^2\), \(b^2\), and \(c^2\) in terms of the sine of the angles.

### Step 10: Establish the relationship
From the derived equation, we find that:

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

This indicates that \(a^2\), \(b^2\), and \(c^2\) are in an arithmetic progression (AP).

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

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