Let `alpha, beta` be such that `pi lt alpha-betalt3pi` if `sin alpha+sinbeta` = `-21/65` and `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 step by step, we will start with the given equations and manipulate them to find the desired value of \( \cos(\alpha - \beta)/2 \).

### Step 1: Write down the given equations
We are given:
\[ \sin \alpha + \sin \beta = -\frac{21}{65} \]
\[ \cos \alpha + \cos \beta = -\frac{27}{65} \]

### Step 2: Use sum-to-product identities
Using the sum-to-product identities, we can express the sums of sine and cosine:
\[ \sin \alpha + \sin \beta = 2 \sin\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right) \]
\[ \cos \alpha + \cos \beta = 2 \cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right) \]

### Step 3: Set up 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 4: Square both equations and add them
Squaring both equations and adding them gives:
\[ \left(2 \sin\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right)\right)^2 + \left(2 \cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right)\right)^2 = \left(-\frac{21}{65}\right)^2 + \left(-\frac{27}{65}\right)^2 \]

### Step 5: Simplify the left side
The left side simplifies to:
\[ 4 \cos^2\left(\frac{\alpha - \beta}{2}\right) \left(\sin^2\left(\frac{\alpha + \beta}{2}\right) + \cos^2\left(\frac{\alpha + \beta}{2}\right)\right) = 4 \cos^2\left(\frac{\alpha - \beta}{2}\right) \]
since \( \sin^2 x + \cos^2 x = 1 \).

### Step 6: Calculate the right side
Calculating the right side:
\[ \left(-\frac{21}{65}\right)^2 + \left(-\frac{27}{65}\right)^2 = \frac{441}{4225} + \frac{729}{4225} = \frac{1170}{4225} \]

### Step 7: Set the equations equal
Setting the left side equal to the right side gives:
\[ 4 \cos^2\left(\frac{\alpha - \beta}{2}\right) = \frac{1170}{4225} \]

### Step 8: Solve for \( \cos^2\left(\frac{\alpha - \beta}{2}\right) \)
Dividing both sides by 4:
\[ \cos^2\left(\frac{\alpha - \beta}{2}\right) = \frac{1170}{16900} \]

### Step 9: Take the square root
Taking the square root gives:
\[ \cos\left(\frac{\alpha - \beta}{2}\right) = \pm \sqrt{\frac{1170}{16900}} = \pm \frac{\sqrt{1170}}{130} \]

### Step 10: Final answer
Thus, the value of \( \cos\left(\frac{\alpha - \beta}{2}\right) \) is:
\[ \cos\left(\frac{\alpha - \beta}{2}\right) = -\frac{\sqrt{1170}}{130} \]