If `cosA=(3)/(4)`, then what is the value of `sin((A)/(2))sin((3A)/(2))`?

Options

A. `(5)/(8)`
B. `(5)/(16)`
C. `(5)/(24)`
D. `(7)/(32)`

Explanation

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

### Step-by-Step Solution:

1. **Use the Product-to-Sum Identity:**
We can use the identity for the product of sines:
\[ 2 \sin A \sin B = \cos(A - B) - \cos(A + B) \]
For our case, let \( A = \frac{A}{2} \) and \( B = \frac{3A}{2} \). Thus,
\[ 2 \sin\left(\frac{A}{2}\right) \sin\left(\frac{3A}{2}\right) = \cos\left(\frac{A}{2} - \frac{3A}{2}\right) - \cos\left(\frac{A}{2} + \frac{3A}{2}\right) \]
This simplifies to:
\[ 2 \sin\left(\frac{A}{2}\right) \sin\left(\frac{3A}{2}\right) = \cos\left(-A\right) - \cos\left(2A\right) \]
Since \( \cos(-A) = \cos A \), we have:
\[ 2 \sin\left(\frac{A}{2}\right) \sin\left(\frac{3A}{2}\right) = \cos A - \cos(2A) \]

2. **Substituting Known Values:**
We know \( \cos A = \frac{3}{4} \). Next, we need to find \( \cos(2A) \) using the double angle formula:
\[ \cos(2A) = 2\cos^2(A) - 1 \]
First, calculate \( \cos^2(A) \):
\[ \cos^2(A) = \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]
Now substitute this into the double angle formula:
\[ \cos(2A) = 2 \cdot \frac{9}{16} - 1 = \frac{18}{16} - \frac{16}{16} = \frac{2}{16} = \frac{1}{8} \]

3. **Substituting Back:**
Now substitute \( \cos A \) and \( \cos(2A) \) back into our equation:
\[ 2 \sin\left(\frac{A}{2}\right) \sin\left(\frac{3A}{2}\right) = \frac{3}{4} - \frac{1}{8} \]
To perform the subtraction, convert \( \frac{3}{4} \) to a fraction with a denominator of 8:
\[ \frac{3}{4} = \frac{6}{8} \]
Thus,
\[ 2 \sin\left(\frac{A}{2}\right) \sin\left(\frac{3A}{2}\right) = \frac{6}{8} - \frac{1}{8} = \frac{5}{8} \]

4. **Final Calculation:**
Now divide both sides by 2 to find \( \sin\left(\frac{A}{2}\right) \sin\left(\frac{3A}{2}\right) \):
\[ \sin\left(\frac{A}{2}\right) \sin\left(\frac{3A}{2}\right) = \frac{5}{8} \cdot \frac{1}{2} = \frac{5}{16} \]

### Final Answer:
\[ \sin\left(\frac{A}{2}\right) \sin\left(\frac{3A}{2}\right) = \frac{5}{16} \]