If `alpha+gamma=2beta` then the expression `(sin alpha-sin gamma)/(cos gamma-cos alpha)` simplifies to:

Options

A. `tan beta`
B. `-tan beta`
C. `cot beta`
D. `-cot beta`

Answer

To simplify the expression \(\frac{\sin \alpha - \sin \gamma}{\cos \gamma - \cos \alpha}\) given that \(\alpha + \gamma = 2\beta\), we can follow these steps:

### Step-by-Step Solution:

1. **Use the Sine Difference Identity**:
The difference of sines can be expressed using the identity:
\[ \sin A - \sin B = 2 \sin\left(\frac{A - B}{2}\right) \cos\left(\frac{A + B}{2}\right) \]
Here, let \(A = \alpha\) and \(B = \gamma\):
\[ \sin \alpha - \sin \gamma = 2 \sin\left(\frac{\alpha - \gamma}{2}\right) \cos\left(\frac{\alpha + \gamma}{2}\right) \]

2. **Use the Cosine Difference Identity**:
The difference of cosines can be expressed using the identity:
\[ \cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \]
Again, let \(A = \gamma\) and \(B = \alpha\):
\[ \cos \gamma - \cos \alpha = -2 \sin\left(\frac{\gamma + \alpha}{2}\right) \sin\left(\frac{\gamma - \alpha}{2}\right) \]

3. **Substituting into the Expression**:
Substitute the identities into the original expression:
\[ \frac{\sin \alpha - \sin \gamma}{\cos \gamma - \cos \alpha} = \frac{2 \sin\left(\frac{\alpha - \gamma}{2}\right) \cos\left(\frac{\alpha + \gamma}{2}\right)}{-2 \sin\left(\frac{\gamma + \alpha}{2}\right) \sin\left(\frac{\gamma - \alpha}{2}\right)} \]
Simplifying this gives:
\[ = -\frac{\sin\left(\frac{\alpha - \gamma}{2}\right) \cos\left(\frac{\alpha + \gamma}{2}\right)}{\sin\left(\frac{\gamma + \alpha}{2}\right) \sin\left(\frac{\gamma - \alpha}{2}\right)} \]

4. **Using the Given Condition**:
Since we know that \(\alpha + \gamma = 2\beta\), we can replace \(\frac{\alpha + \gamma}{2}\) with \(\beta\):
\[ = -\frac{\sin\left(\frac{\alpha - \gamma}{2}\right) \cos(\beta)}{\sin(\beta) \sin\left(\frac{\gamma - \alpha}{2}\right)} \]

5. **Final Simplification**:
Notice that \(\frac{\sin\left(\frac{\alpha - \gamma}{2}\right)}{\sin\left(\frac{\gamma - \alpha}{2}\right)} = -1\) (since \(\frac{\alpha - \gamma}{2} = -\frac{\gamma - \alpha}{2}\)):
\[ = \frac{\cos(\beta)}{\sin(\beta)} = \cot(\beta) \]

Thus, the expression simplifies to:
\[ \cot(\beta) \]