Let `alpha, beta` be such that `pi lt alph-betalt3piif sin alpha+sinbeta=-21/65and cos alpha+cos beta =-27/65,` then the value of
`cos""(alpha-beta)/(2),` is

Options

A. `-(6)/(65)`
B. `(3)/(sqrt130)`
C. `6/65`
D. `-(3)/(sqrt130)`

Explanation

To solve the problem, we start with the given equations:

1. \( \sin \alpha + \sin \beta = -\frac{21}{65} \)
2. \( \cos \alpha + \cos \beta = -\frac{27}{65} \)

We need to find the value of \( \cos\left(\frac{\alpha - \beta}{2}\right) \).

### Step 1: Use the sum-to-product identities
Using the sum-to-product identities for sine and cosine, we can express the sums in terms of products:

\[ \sin \alpha + \sin \beta = 2 \sin\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right) \]

\[ \cos \alpha + \cos \beta = 2 \cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right) \]

### Step 2: Set up the equations
From the identities, we can set up the following equations:

\[ 2 \sin\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right) = -\frac{21}{65} \quad (1) \]

\[ 2 \cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right) = -\frac{27}{65} \quad (2) \]

### Step 3: Square and add the equations
Next, we square both equations and add them:

\[ \left(2 \sin\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right)\right)^2 + \left(2 \cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right)\right)^2 = \left(-\frac{21}{65}\right)^2 + \left(-\frac{27}{65}\right)^2 \]

This simplifies to:

\[ 4 \cos^2\left(\frac{\alpha - \beta}{2}\right) \left(\sin^2\left(\frac{\alpha + \beta}{2}\right) + \cos^2\left(\frac{\alpha + \beta}{2}\right)\right) = \frac{441}{4225} + \frac{729}{4225} \]

### Step 4: Simplify using the Pythagorean identity
Using the identity \( \sin^2 x + \cos^2 x = 1 \):

\[ 4 \cos^2\left(\frac{\alpha - \beta}{2}\right) = \frac{441 + 729}{4225} = \frac{1170}{4225} \]

### Step 5: Solve for \( \cos^2\left(\frac{\alpha - \beta}{2}\right) \)
Now we can isolate \( \cos^2\left(\frac{\alpha - \beta}{2}\right) \):

\[ \cos^2\left(\frac{\alpha - \beta}{2}\right) = \frac{1170}{4 \times 4225} = \frac{1170}{16900} \]

### Step 6: Take the square root
Taking the square root gives us:

\[ \cos\left(\frac{\alpha - \beta}{2}\right) = \pm \sqrt{\frac{1170}{16900}} = \pm \frac{\sqrt{1170}}{130} \]

### Step 7: Determine the sign
Given that \( \alpha - \beta \) is between \( \pi \) and \( 3\pi \), \( \frac{\alpha - \beta}{2} \) is between \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \), where cosine is negative. Therefore, we take the negative value:

\[ \cos\left(\frac{\alpha - \beta}{2}\right) = -\frac{\sqrt{1170}}{130} \]

### Final Answer
Thus, the value of \( \cos\left(\frac{\alpha - \beta}{2}\right) \) is:

\[ \cos\left(\frac{\alpha - \beta}{2}\right) = -\frac{\sqrt{1170}}{130} \]