`5sintheta.sin8theta=`

Options

A. `(3)/(2)(cos7theta-costheta)`
B. `(5)/(2)(cos9theta-cos7theta)`
C. `(5)/(2)(cos7theta-cos9theta)`
D. none of these

Answer

To solve the expression \( 5 \sin \theta \sin 8\theta \), we will use the trigonometric identity for the product of sines. The identity states:

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

### Step-by-Step Solution:

1. **Identify A and B**:
In our case, let \( A = \theta \) and \( B = 8\theta \).

2. **Multiply and Divide by 2**:
To use the identity, we can rewrite the expression:
\[ 5 \sin \theta \sin 8\theta = \frac{5}{2} \cdot 2 \sin \theta \sin 8\theta \]

3. **Apply the Identity**:
Now apply the identity:
\[ 2 \sin \theta \sin 8\theta = \cos(\theta - 8\theta) - \cos(\theta + 8\theta) \]
This simplifies to:
\[ 2 \sin \theta \sin 8\theta = \cos(-7\theta) - \cos(9\theta) \]

4. **Use the Property of Cosine**:
Since \( \cos(-x) = \cos(x) \), we can simplify further:
\[ 2 \sin \theta \sin 8\theta = \cos(7\theta) - \cos(9\theta) \]

5. **Substitute Back**:
Now substitute this back into our expression:
\[ 5 \sin \theta \sin 8\theta = \frac{5}{2} (\cos(7\theta) - \cos(9\theta)) \]

6. **Final Result**:
Thus, the final result is:
\[ 5 \sin \theta \sin 8\theta = \frac{5}{2} \cos(7\theta) - \frac{5}{2} \cos(9\theta) \]

### Final Answer:
\[ 5 \sin \theta \sin 8\theta = \frac{5}{2} (\cos(7\theta) - \cos(9\theta)) \]