Let `alpha, beta` be such that `pi lt alpha - beta lt 3pi ` . If `sin alpha + sin beta = -(21)/( 65)` and `cos alpha + cos beta = - ( 27)/( 65)` then the value of `cos "" ( alpha - beta)/( 2)` is

Options

A. `- (3)/( sqrt( 130))`
B. `( 3)/( sqrt( 130))`
C. `( 6)/( 65)`
D. `-( 6)/( 65)`

Answer

To solve the problem step by step, let's break it down as follows:

### Step 1: Given Equations
We are given:
1. \( \sin \alpha + \sin \beta = -\frac{21}{65} \)
2. \( \cos \alpha + \cos \beta = -\frac{27}{65} \)

### Step 2: Square Both Equations
We will square both equations to utilize the Pythagorean identity.

**Squaring the first equation:**
\[ (\sin \alpha + \sin \beta)^2 = \left(-\frac{21}{65}\right)^2 \]
This expands to:
\[ \sin^2 \alpha + \sin^2 \beta + 2 \sin \alpha \sin \beta = \frac{441}{4225} \]

**Squaring the second equation:**
\[ (\cos \alpha + \cos \beta)^2 = \left(-\frac{27}{65}\right)^2 \]
This expands to:
\[ \cos^2 \alpha + \cos^2 \beta + 2 \cos \alpha \cos \beta = \frac{729}{4225} \]

### Step 3: Add Both Equations
Now, we add both squared equations:
\[ \sin^2 \alpha + \sin^2 \beta + 2 \sin \alpha \sin \beta + \cos^2 \alpha + \cos^2 \beta + 2 \cos \alpha \cos \beta = \frac{441}{4225} + \frac{729}{4225} \]

Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ 1 + 1 + 2(\sin \alpha \sin \beta + \cos \alpha \cos \beta) = \frac{1170}{4225} \]
This simplifies to:
\[ 2 + 2(\sin \alpha \sin \beta + \cos \alpha \cos \beta) = \frac{1170}{4225} \]

### Step 4: Simplify the Equation
Now, isolate the term with sine and cosine:
\[ 2(\sin \alpha \sin \beta + \cos \alpha \cos \beta) = \frac{1170}{4225} - 2 \]
Convert 2 into a fraction with a common denominator:
\[ 2 = \frac{8450}{4225} \]
Thus:
\[ 2(\sin \alpha \sin \beta + \cos \alpha \cos \beta) = \frac{1170 - 8450}{4225} = \frac{-7280}{4225} \]
Divide by 2:
\[ \sin \alpha \sin \beta + \cos \alpha \cos \beta = \frac{-3640}{4225} \]

### Step 5: Use Cosine of Difference Identity
Using the identity \( \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \):
\[ \cos(\alpha - \beta) = \frac{-3640}{4225} \]

### Step 6: Find Cosine Half Angle
Now, we need to find \( \cos\left(\frac{\alpha - \beta}{2}\right) \).
Using the half-angle identity:
\[ \cos(\alpha - \beta) = 2 \cos^2\left(\frac{\alpha - \beta}{2}\right) - 1 \]
Set this equal to our previous result:
\[ \frac{-3640}{4225} = 2 \cos^2\left(\frac{\alpha - \beta}{2}\right) - 1 \]
Rearranging gives:
\[ 2 \cos^2\left(\frac{\alpha - \beta}{2}\right) = \frac{-3640}{4225} + 1 \]
Convert 1 into a fraction:
\[ 1 = \frac{4225}{4225} \]
Thus:
\[ 2 \cos^2\left(\frac{\alpha - \beta}{2}\right) = \frac{4225 - 3640}{4225} = \frac{585}{4225} \]
Now divide by 2:
\[ \cos^2\left(\frac{\alpha - \beta}{2}\right) = \frac{585}{8450} \]

### Step 7: Take Square Root
Taking the square root:
\[ \cos\left(\frac{\alpha - \beta}{2}\right) = \pm \sqrt{\frac{585}{8450}} \]
Since \( \alpha - \beta \) is between \( \pi \) and \( 3\pi \), the cosine value will be negative:
\[ \cos\left(\frac{\alpha - \beta}{2}\right) = -\sqrt{\frac{585}{8450}} \]

### Final Answer
Thus, the value of \( \cos\left(\frac{\alpha - \beta}{2}\right) \) is:
\[ \cos\left(\frac{\alpha - \beta}{2}\right) = -\frac{3}{\sqrt{130}} \]