If cosA=3/4, then 32 sin (A/2) sin ((5A) /2)= ------------- (A) √11 (B) -√11 (C) 11 (D) -11

Explanation

To solve the problem, we need to find the value of \( 32 \sin\left(\frac{A}{2}\right) \sin\left(\frac{5A}{2}\right) \) given that \( \cos A = \frac{3}{4} \).

### Step-by-Step Solution:

1. **Use the Double Angle Identity**:
We can use the identity for the product of sines:
\[ 2 \sin x \sin y = \cos(x-y) - \cos(x+y) \]
Here, we will let \( x = \frac{A}{2} \) and \( y = \frac{5A}{2} \).

2. **Rewrite the Expression**:
Thus, we can rewrite:
\[ 32 \sin\left(\frac{A}{2}\right) \sin\left(\frac{5A}{2}\right) = 16 \left(2 \sin\left(\frac{A}{2}\right) \sin\left(\frac{5A}{2}\right)\right) = 16 \left(\cos\left(\frac{A}{2} - \frac{5A}{2}\right) - \cos\left(\frac{A}{2} + \frac{5A}{2}\right)\right) \]

3. **Simplify the Angles**:
Simplifying the angles:
\[ \frac{A}{2} - \frac{5A}{2} = -2A \quad \text{and} \quad \frac{A}{2} + \frac{5A}{2} = 3A \]
Therefore, we have:
\[ 32 \sin\left(\frac{A}{2}\right) \sin\left(\frac{5A}{2}\right) = 16 \left(\cos(-2A) - \cos(3A)\right) \]
Since \( \cos(-x) = \cos(x) \), this simplifies to:
\[ 16 \left(\cos(2A) - \cos(3A)\right) \]

4. **Use Cosine Double Angle Formula**:
We know:
\[ \cos(2A) = 2\cos^2(A) - 1 \quad \text{and} \quad \cos(3A) = 4\cos^3(A) - 3\cos(A) \]
Substituting \( \cos A = \frac{3}{4} \):
\[ \cos(2A) = 2\left(\frac{3}{4}\right)^2 - 1 = 2 \cdot \frac{9}{16} - 1 = \frac{18}{16} - \frac{16}{16} = \frac{2}{16} = \frac{1}{8} \]
\[ \cos(3A) = 4\left(\frac{3}{4}\right)^3 - 3\left(\frac{3}{4}\right) = 4 \cdot \frac{27}{64} - \frac{9}{4} = \frac{108}{64} - \frac{144}{64} = -\frac{36}{64} = -\frac{9}{16} \]

5. **Substitute Back**:
Now substituting these values back:
\[ 32 \sin\left(\frac{A}{2}\right) \sin\left(\frac{5A}{2}\right) = 16 \left(\frac{1}{8} - \left(-\frac{9}{16}\right)\right) = 16 \left(\frac{1}{8} + \frac{9}{16}\right) \]
To add these fractions, we need a common denominator:
\[ \frac{1}{8} = \frac{2}{16} \quad \text{so} \quad \frac{1}{8} + \frac{9}{16} = \frac{2}{16} + \frac{9}{16} = \frac{11}{16} \]
Thus,
\[ 32 \sin\left(\frac{A}{2}\right) \sin\left(\frac{5A}{2}\right) = 16 \cdot \frac{11}{16} = 11 \]

### Final Answer:
The value is \( 11 \).