The value of `((cos alpha+cos beta)/(sin alpha-sin beta))^(n)+((sin alpha+sin beta)/(cos alpha-cos beta))^(n)` (where n is a whole number) is equal to

Options

A. 0, when n is odd
B. `2tan^(n)(alpha-beta)/(2), AA n`
C. `2cot^(n)""(alpha-beta)/(2)`, when n is even
D. `2cot^(n)""(alpha+beta)/(2)`, when n is even

Answer

To solve the expression

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

where \( n \) is a whole number, we will simplify each term step by step.

### Step 1: Simplify the first term

Using the trigonometric identities, we can rewrite \(\cos \alpha + \cos \beta\) and \(\sin \alpha - \sin \beta\):

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

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

Thus, we can rewrite the first term:

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

Cancelling \(2 \cos\left(\frac{\alpha - \beta}{2}\right)\) (assuming \(\cos\left(\frac{\alpha - \beta}{2}\right) \neq 0\)), we get:

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

### Step 2: Simplify the second term

Now, for the second term:

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

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

Thus, we can rewrite the second term:

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

Cancelling \(2 \sin\left(\frac{\alpha + \beta}{2}\right)\) (assuming \(\sin\left(\frac{\alpha + \beta}{2}\right) \neq 0\)), we get:

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

### Step 3: Combine the terms

Now, we can combine the two simplified terms:

\[ \left(\cot\left(\frac{\alpha + \beta}{2}\right)\right)^{n} + \left(-\cot\left(\frac{\alpha - \beta}{2}\right)\right)^{n} \]

### Step 4: Analyze based on the parity of \( n \)

1. **If \( n \) is odd**:
\[ (-\cot\left(\frac{\alpha - \beta}{2}\right))^{n} = -\cot\left(\frac{\alpha - \beta}{2}\right)^{n} \]
Thus, the expression becomes:
\[ \cot\left(\frac{\alpha + \beta}{2}\right)^{n} - \cot\left(\frac{\alpha - \beta}{2}\right)^{n} = 0 \]

2. **If \( n \) is even**:
\[ (-\cot\left(\frac{\alpha - \beta}{2}\right))^{n} = \cot\left(\frac{\alpha - \beta}{2}\right)^{n} \]
Thus, the expression becomes:
\[ \cot\left(\frac{\alpha + \beta}{2}\right)^{n} + \cot\left(\frac{\alpha - \beta}{2}\right)^{n} = 2 \cot\left(\frac{\alpha - \beta}{2}\right)^{n} \]

### Final Result

- If \( n \) is odd, the value is \( 0 \).
- If \( n \) is even, the value is \( 2 \cot\left(\frac{\alpha - \beta}{2}\right)^{n} \).