If `alpha+beta+gamma=2 theta`,then `cos theta + cos(theta - alpha) + cos(theta - beta) + cos(theta - gamma)` =

Options

A. `4sin""(alpha)/(2)sin""(beta)/(2)sin""(gamma)/(2)
B. `4cos""(alpha)/(2)cos""(beta)/(2)cos""(gamma)/(2)
C. `4sin""(alpha)/(2)sin""(beta)/(2)sin""(gamma)/(2)
D. `4sinalphasinbetasingamma`

Answer

To solve the problem, we need to evaluate the expression \( \cos \theta + \cos(\theta - \alpha) + \cos(\theta - \beta) + \cos(\theta - \gamma) \) given that \( \alpha + \beta + \gamma = 2\theta \).

### Step-by-Step Solution:

1. **Start with the given expression:**
\[ E = \cos \theta + \cos(\theta - \alpha) + \cos(\theta - \beta) + \cos(\theta - \gamma) \]

2. **Use the cosine addition formula:**
We can group the terms in pairs and apply the cosine addition formula:
\[ \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]

3. **Group the first two terms:**
Let's first group \( \cos \theta \) and \( \cos(\theta - \alpha) \):
\[ \cos \theta + \cos(\theta - \alpha) = 2 \cos\left(\frac{\theta + (\theta - \alpha)}{2}\right) \cos\left(\frac{\theta - (\theta - \alpha)}{2}\right) \]
Simplifying this gives:
\[ = 2 \cos\left(\theta - \frac{\alpha}{2}\right) \cos\left(\frac{\alpha}{2}\right) \]

4. **Group the next two terms:**
Now group \( \cos(\theta - \beta) \) and \( \cos(\theta - \gamma) \):
\[ \cos(\theta - \beta) + \cos(\theta - \gamma) = 2 \cos\left(\frac{(\theta - \beta) + (\theta - \gamma)}{2}\right) \cos\left(\frac{(\theta - \beta) - (\theta - \gamma)}{2}\right) \]
Simplifying this gives:
\[ = 2 \cos\left(\theta - \frac{\beta + \gamma}{2}\right) \cos\left(\frac{\gamma - \beta}{2}\right) \]

5. **Combine the results:**
Now we can combine the results from steps 3 and 4:
\[ E = 2 \cos\left(\theta - \frac{\alpha}{2}\right) \cos\left(\frac{\alpha}{2}\right) + 2 \cos\left(\theta - \frac{\beta + \gamma}{2}\right) \cos\left(\frac{\gamma - \beta}{2}\right) \]

6. **Use the identity \( \alpha + \beta + \gamma = 2\theta \):**
Substitute \( \beta + \gamma = 2\theta - \alpha \):
\[ E = 2 \cos\left(\theta - \frac{\alpha}{2}\right) \cos\left(\frac{\alpha}{2}\right) + 2 \cos\left(\theta - \frac{2\theta - \alpha}{2}\right) \cos\left(\frac{\gamma - \beta}{2}\right) \]

7. **Final simplification:**
After simplification, we find:
\[ E = 4 \cos\left(\frac{\alpha}{2}\right) \cos\left(\frac{\beta}{2}\right) \cos\left(\frac{\gamma}{2}\right) \]

### Final Result:
Thus, the value of the expression \( \cos \theta + \cos(\theta - \alpha) + \cos(\theta - \beta) + \cos(\theta - \gamma) \) is:
\[ \boxed{4 \cos\left(\frac{\alpha}{2}\right) \cos\left(\frac{\beta}{2}\right) \cos\left(\frac{\gamma}{2}\right)} \]