If `sin^(2)A=x` and `underset(r=1)overset(4)Pi sin (r A)=ax^(2)+bx^(3)+cx^(4)+dx^(5)`, then the value of `10a-7b+15c-5d` must be

Answer

To solve the problem, we start with the given information and follow a systematic approach.

### Step 1: Understand the given information
We are given that \( \sin^2 A = x \) and we need to evaluate the expression \( \underset{r=1}{\overset{4}{\prod}} \sin(rA) = ax^2 + bx^3 + cx^4 + dx^5 \).

### Step 2: Rewrite the product
The product \( \prod_{r=1}^{4} \sin(rA) \) can be expanded as:
\[ \sin(A) \cdot \sin(2A) \cdot \sin(3A) \cdot \sin(4A) \]

### Step 3: Use trigonometric identities
Using the double angle and triple angle formulas:
- \( \sin(2A) = 2 \sin(A) \cos(A) \)
- \( \sin(3A) = 3 \sin(A) - 4 \sin^3(A) \)
- \( \sin(4A) = 2 \sin(2A) \cos(2A) = 2(2 \sin(A) \cos(A))(1 - 2 \sin^2(A)) \)

### Step 4: Substitute and simplify
Substituting these identities into the product:
\[ \sin(A) \cdot (2 \sin(A) \cos(A)) \cdot (3 \sin(A) - 4 \sin^3(A)) \cdot [2(2 \sin(A) \cos(A))(1 - 2 \sin^2(A))] \]

### Step 5: Combine terms
This can be simplified step by step:
1. Combine \( \sin(A) \) terms.
2. Use \( \sin^2(A) = x \) to express everything in terms of \( x \).

### Step 6: Expand the expression
After substituting \( \sin^2(A) = x \), we will have a polynomial in \( x \):
\[ \prod_{r=1}^{4} \sin(rA) = ax^2 + bx^3 + cx^4 + dx^5 \]

### Step 7: Identify coefficients
By comparing coefficients from the expanded polynomial with \( ax^2 + bx^3 + cx^4 + dx^5 \), we find:
- \( a = 24 \)
- \( b = -104 \)
- \( c = 144 \)
- \( d = -64 \)

### Step 8: Calculate the final expression
Now we need to calculate \( 10a - 7b + 15c - 5d \):
\[ 10a = 10 \times 24 = 240 \]
\[ -7b = -7 \times (-104) = 728 \]
\[ 15c = 15 \times 144 = 2160 \]
\[ -5d = -5 \times (-64) = 320 \]

### Step 9: Combine all terms
Now, combine all these results:
\[ 10a - 7b + 15c - 5d = 240 + 728 + 2160 + 320 = 3448 \]

### Final Answer
Thus, the value of \( 10a - 7b + 15c - 5d \) is \( 3448 \).

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