If `2 sin alphacos beta sin gamma=sinbeta sin(alpha+gamma),then tan alpha,tan beta and gamma` are in

Options

A. A.P.
B. G.P.
C. H.P.
D. none of these

Explanation

To solve the problem, we need to analyze the given equation and determine the relationship between \( \tan \alpha \), \( \tan \beta \), and \( \tan \gamma \).

### Step-by-Step Solution:

1. **Start with the Given Equation:**
\[ 2 \sin \alpha \cos \beta \sin \gamma = \sin \beta \sin(\alpha + \gamma) \]

2. **Use the Sine Addition Formula:**
Recall that \( \sin(\alpha + \gamma) = \sin \alpha \cos \gamma + \cos \alpha \sin \gamma \). Substitute this into the equation:
\[ 2 \sin \alpha \cos \beta \sin \gamma = \sin \beta (\sin \alpha \cos \gamma + \cos \alpha \sin \gamma) \]

3. **Distribute \( \sin \beta \):**
\[ 2 \sin \alpha \cos \beta \sin \gamma = \sin \beta \sin \alpha \cos \gamma + \sin \beta \cos \alpha \sin \gamma \]

4. **Rearrange the Equation:**
Rearranging gives:
\[ 2 \sin \alpha \cos \beta \sin \gamma - \sin \beta \sin \alpha \cos \gamma - \sin \beta \cos \alpha \sin \gamma = 0 \]

5. **Factor Out Common Terms:**
Group the terms involving \( \sin \gamma \):
\[ \sin \gamma (2 \sin \alpha \cos \beta - \sin \beta \cos \alpha) - \sin \beta \sin \alpha \cos \gamma = 0 \]

6. **Set Each Factor to Zero:**
For the equation to hold, either \( \sin \gamma = 0 \) or:
\[ 2 \sin \alpha \cos \beta - \sin \beta \cos \alpha = \frac{\sin \beta \sin \alpha \cos \gamma}{\sin \gamma} \]

7. **Divide by \( \sin \alpha \sin \gamma \) (assuming they are non-zero):**
\[ \frac{2 \cos \beta}{\sin \gamma} = \frac{\sin \beta \cos \alpha}{\sin \alpha} \]

8. **Express in Terms of Cotangent:**
Rearranging gives:
\[ 2 \cot \beta = \cot \alpha + \cot \gamma \]

9. **Identify the Relationship:**
The equation \( 2 \cot \beta = \cot \alpha + \cot \gamma \) indicates that \( \cot \alpha, \cot \beta, \cot \gamma \) are in Arithmetic Progression (AP).

10. **Conclude the Relationship of Tangents:**
If \( \cot \alpha, \cot \beta, \cot \gamma \) are in AP, then \( \tan \alpha, \tan \beta, \tan \gamma \) are in Harmonic Progression (HP).

### Final Answer:
Thus, \( \tan \alpha, \tan \beta, \tan \gamma \) are in Harmonic Progression (HP).

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