If A,B,C,D are angles of a cyclic quadrilateral, then the value of `cosA+cosB+cosC+cosD` is

Options

A. `-1`
B. 1
C. 0
D. 2

Answer

To solve the problem, we need to find the value of \( \cos A + \cos B + \cos C + \cos D \) where \( A, B, C, D \) are the angles of a cyclic quadrilateral.

### Step-by-step Solution:

1. **Understanding Cyclic Quadrilateral:**
A cyclic quadrilateral is a quadrilateral whose vertices lie on a circle. One important property of cyclic quadrilaterals is that opposite angles are supplementary.

2. **Using the Property of Supplementary Angles:**
Since \( A \) and \( C \) are opposite angles, we have:
\[ A + C = 180^\circ \quad \text{(or } \pi \text{ radians)} \]
Similarly, for angles \( B \) and \( D \):
\[ B + D = 180^\circ \quad \text{(or } \pi \text{ radians)} \]

3. **Expressing One Angle in Terms of Another:**
From the first equation, we can express \( C \) in terms of \( A \):
\[ C = 180^\circ - A \quad \text{(or } C = \pi - A \text{)} \]
From the second equation, we can express \( D \) in terms of \( B \):
\[ D = 180^\circ - B \quad \text{(or } D = \pi - B \text{)} \]

4. **Applying the Cosine Function:**
We can now apply the cosine function to both expressions:
\[ \cos C = \cos(180^\circ - A) = -\cos A \]
\[ \cos D = \cos(180^\circ - B) = -\cos B \]

5. **Substituting Back into the Expression:**
Now substituting these results back into our original expression:
\[ \cos A + \cos B + \cos C + \cos D = \cos A + \cos B - \cos A - \cos B \]

6. **Simplifying the Expression:**
When we simplify the above expression, we get:
\[ \cos A + \cos B - \cos A - \cos B = 0 \]

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