What is the value of `(cos x + cosy)/(sinx + siny)`

Options

A. `tan"" ((x+y)/2)`
B. `tan"" ((x+y)/2)`
C. `cot"" ((x + y)/2)`
D. `cot "" ((x+y)/2)`

Answer

To find the value of \((\cos x + \cos y) / (\sin x + \sin y)\), we can use trigonometric identities.

### Step-by-step Solution:

1. **Use the Sum-to-Product Identities**:
We start by applying the sum-to-product identities for cosine and sine:
\[ \cos x + \cos y = 2 \cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) \]
\[ \sin x + \sin y = 2 \sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) \]

2. **Substituting the Identities**:
Substitute these identities into the original expression:
\[ \frac{\cos x + \cos y}{\sin x + \sin y} = \frac{2 \cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right)}{2 \sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right)} \]

3. **Canceling Common Factors**:
Notice that \(2 \cos\left(\frac{x-y}{2}\right)\) appears in both the numerator and the denominator. As long as \(\cos\left(\frac{x-y}{2}\right) \neq 0\), we can cancel it:
\[ = \frac{\cos\left(\frac{x+y}{2}\right)}{\sin\left(\frac{x+y}{2}\right)} \]

4. **Using the Definition of Cotangent**:
The expression \(\frac{\cos\left(\frac{x+y}{2}\right)}{\sin\left(\frac{x+y}{2}\right)}\) is the definition of cotangent:
\[ = \cot\left(\frac{x+y}{2}\right) \]

### Final Answer:
Thus, the value of \(\frac{\cos x + \cos y}{\sin x + \sin y}\) is:
\[ \cot\left(\frac{x+y}{2}\right) \]