If ABCD is a cyclic quadrilateral, then the value of cosA-cosB+cosC-cosD is equal to

Options

A. 1
B. 0
C. `-1`
D. none of these

Answer

To solve the problem, we need to find the value of the expression \( \cos A - \cos B + \cos C - \cos D \) given that \( ABCD \) is a cyclic quadrilateral.

### Step-by-Step Solution:

1. **Understanding Cyclic Quadrilaterals**:
In a cyclic quadrilateral, the sum of the opposite angles is equal to \( 180^\circ \). Therefore, we have:
\[ A + C = 180^\circ \quad \text{and} \quad B + D = 180^\circ \]

2. **Expressing Angles**:
From the above relationships, we can express \( C \) and \( D \) in terms of \( A \) and \( B \):
\[ C = 180^\circ - A \quad \text{and} \quad D = 180^\circ - B \]

3. **Substituting in the Expression**:
Now we substitute \( C \) and \( D \) into the expression \( \cos A - \cos B + \cos C - \cos D \):
\[ \cos A - \cos B + \cos(180^\circ - A) - \cos(180^\circ - B) \]

4. **Using Cosine Properties**:
We know that \( \cos(180^\circ - \theta) = -\cos(\theta) \). Therefore, we can rewrite the expression:
\[ \cos A - \cos B - \cos A + \cos B \]

5. **Simplifying the Expression**:
Now, we can simplify the expression:
\[ \cos A - \cos A + \cos B - \cos B = 0 \]

6. **Final Result**:
Thus, the value of the expression \( \cos A - \cos B + \cos C - \cos D \) is:
\[ \boxed{0} \]