If `alpha, beta, gamma,delta` be four angles of a cyclic quadrilateral taken in clockwise direction then the value of `(2+Sigma cos alpha cos beta)` will be:

Options

A. `sin^(2)alpha+sin^(2)beta`
B. `cos^(2)gamma+cos^(2)delta`
C. `sin^(2)alpha +sin^(2)delta`
D. `cos^(2)beta+cos^(2)gamma`

Answer

To solve the problem, we need to find the value of \(2 + \Sigma \cos \alpha \cos \beta\) for a cyclic quadrilateral with angles \(\alpha, \beta, \gamma, \delta\).

### Step-by-Step Solution:

1. **Understanding the Properties of Cyclic Quadrilaterals**:
In a cyclic quadrilateral, the sum of opposite angles is \(180^\circ\). Therefore, we have:
\[ \alpha + \gamma = 180^\circ \quad \text{and} \quad \beta + \delta = 180^\circ \]

2. **Expressing Angles**:
From the above properties, we can express \(\gamma\) and \(\delta\) in terms of \(\alpha\) and \(\beta\):
\[ \gamma = 180^\circ - \alpha \quad \text{and} \quad \delta = 180^\circ - \beta \]

3. **Using Cosine Properties**:
We can use the cosine of these angles:
\[ \cos \gamma = \cos(180^\circ - \alpha) = -\cos \alpha \]
\[ \cos \delta = \cos(180^\circ - \beta) = -\cos \beta \]

4. **Expanding the Expression**:
We need to evaluate:
\[ \Sigma \cos \alpha \cos \beta = \cos \alpha \cos \beta + \cos \alpha \cos \gamma + \cos \alpha \cos \delta + \cos \beta \cos \gamma + \cos \beta \cos \delta + \cos \gamma \cos \delta \]

5. **Substituting Values**:
Substitute \(\cos \gamma\) and \(\cos \delta\) into the expression:
\[ \Sigma \cos \alpha \cos \beta = \cos \alpha \cos \beta + \cos \alpha (-\cos \alpha) + \cos \alpha (-\cos \beta) + \cos \beta (-\cos \alpha) + \cos \beta (-\cos \beta) + (-\cos \alpha)(-\cos \beta) \]
This simplifies to:
\[ = \cos \alpha \cos \beta - \cos^2 \alpha - \cos^2 \beta + \cos^2 \alpha + \cos^2 \beta \]
The terms \(-\cos^2 \alpha\) and \(-\cos^2 \beta\) cancel out.

6. **Final Simplification**:
Thus, we find:
\[ \Sigma \cos \alpha \cos \beta = 0 \]

7. **Adding 2**:
Now, we add 2 to our result:
\[ 2 + \Sigma \cos \alpha \cos \beta = 2 + 0 = 2 \]

### Conclusion:
The value of \(2 + \Sigma \cos \alpha \cos \beta\) is \(2\).