`4 sin ((pi)/(3) + theta)sin ((pi)/(3)- theta)=`

Options

A. ` 1+ cos theta `
B. `1- 2 cos 2 theta`
C. `2 cos 2 theta -1`
D. `1+ 2 cos 2 theta`

Answer

To solve the expression \(4 \sin\left(\frac{\pi}{3} + \theta\right) \sin\left(\frac{\pi}{3} - \theta\right)\), we will use the product-to-sum identities in trigonometry.

### Step-by-Step Solution:

1. **Identify the formula**: We will use the product-to-sum formula:
\[ 2 \sin A \sin B = \cos(A - B) - \cos(A + B) \]
Here, let \(A = \frac{\pi}{3} + \theta\) and \(B = \frac{\pi}{3} - \theta\).

2. **Apply the formula**:
\[ 4 \sin\left(\frac{\pi}{3} + \theta\right) \sin\left(\frac{\pi}{3} - \theta\right) = 2 \cdot 2 \sin\left(\frac{\pi}{3} + \theta\right) \sin\left(\frac{\pi}{3} - \theta\right) \]
Using the product-to-sum identity:
\[ = 2 \left[\cos\left(\left(\frac{\pi}{3} + \theta\right) - \left(\frac{\pi}{3} - \theta\right)\right) - \cos\left(\left(\frac{\pi}{3} + \theta\right) + \left(\frac{\pi}{3} - \theta\right)\right)\right] \]

3. **Simplify the angles**:
- For \(A - B\):
\[ \left(\frac{\pi}{3} + \theta\right) - \left(\frac{\pi}{3} - \theta\right) = 2\theta \]
- For \(A + B\):
\[ \left(\frac{\pi}{3} + \theta\right) + \left(\frac{\pi}{3} - \theta\right) = \frac{2\pi}{3} \]

4. **Substituting back into the equation**:
\[ = 2 \left[\cos(2\theta) - \cos\left(\frac{2\pi}{3}\right)\right] \]

5. **Evaluate \(\cos\left(\frac{2\pi}{3}\right)\)**:
\[ \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \]
Therefore:
\[ = 2 \left[\cos(2\theta) - \left(-\frac{1}{2}\right)\right] \]
\[ = 2 \left[\cos(2\theta) + \frac{1}{2}\right] \]

6. **Distribute the 2**:
\[ = 2\cos(2\theta) + 1 \]

### Final Result:
Thus, the expression \(4 \sin\left(\frac{\pi}{3} + \theta\right) \sin\left(\frac{\pi}{3} - \theta\right)\) simplifies to:
\[ \boxed{2\cos(2\theta) + 1} \]