If A,B,C,D are the angles of a cyclic quadrilateral and cos A+cos B= K(cos C + cos D), then the value of K is

Options

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

Answer

To solve the problem, we need to find the value of K in the equation \( \cos A + \cos B = K(\cos C + \cos D) \) given that A, B, C, and D are angles of a cyclic quadrilateral.

### Step-by-Step Solution:

1. **Understand the properties of cyclic quadrilaterals**:
In a cyclic quadrilateral, the sum of opposite angles is 180 degrees. Therefore, we have:
\[ A + C = 180^\circ \quad \text{and} \quad B + D = 180^\circ \]

2. **Express \( \cos A \) and \( \cos B \)**:
Using the property of cosine in relation to angles summing to 180 degrees, we can express \( \cos A \) and \( \cos B \):
\[ \cos A = \cos(180^\circ - C) = -\cos C \]
\[ \cos B = \cos(180^\circ - D) = -\cos D \]

3. **Substitute \( \cos A \) and \( \cos B \) into the equation**:
Now we substitute these expressions into the original equation:
\[ \cos A + \cos B = -\cos C - \cos D \]

4. **Rewrite the equation**:
Thus, we can rewrite the equation as:
\[ -(\cos C + \cos D) = K(\cos C + \cos D) \]

5. **Solve for K**:
To isolate K, we can divide both sides by \( \cos C + \cos D \) (assuming \( \cos C + \cos D \neq 0 \)):
\[ -1 = K \]
Hence, we find that:
\[ K = -1 \]

### Final Answer:
The value of \( K \) is \( -1 \).