If `tan beta=2sin alpha sin gamma co sec(alpha+gamma)`, then `cot alpha,cot beta,cotgamma` are in

Options

A. AP
B. GP
C. HP
D. none of these.

Explanation

To solve the problem, we start with the given equation:

\[ \tan \beta = 2 \sin \alpha \sin \gamma \csc(\alpha + \gamma) \]

### Step 1: Rewrite \(\tan \beta\) in terms of \(\cot \beta\)

We know that:

\[ \tan \beta = \frac{1}{\cot \beta} \]

Thus, we can rewrite the equation as:

\[ \frac{1}{\cot \beta} = 2 \sin \alpha \sin \gamma \csc(\alpha + \gamma) \]

### Step 2: Express \(\cot \beta\)

Taking the reciprocal of both sides gives:

\[ \cot \beta = \frac{1}{2 \sin \alpha \sin \gamma \csc(\alpha + \gamma)} \]

Using the identity \(\csc(\theta) = \frac{1}{\sin(\theta)}\), we can rewrite \(\csc(\alpha + \gamma)\):

\[ \cot \beta = \frac{\sin(\alpha + \gamma)}{2 \sin \alpha \sin \gamma} \]

### Step 3: Expand \(\sin(\alpha + \gamma)\)

Using the sine addition formula:

\[ \sin(\alpha + \gamma) = \sin \alpha \cos \gamma + \cos \alpha \sin \gamma \]

Substituting this back into the expression for \(\cot \beta\):

\[ \cot \beta = \frac{\sin \alpha \cos \gamma + \cos \alpha \sin \gamma}{2 \sin \alpha \sin \gamma} \]

### Step 4: Simplify the expression

We can separate the terms in the numerator:

\[ \cot \beta = \frac{\sin \alpha \cos \gamma}{2 \sin \alpha \sin \gamma} + \frac{\cos \alpha \sin \gamma}{2 \sin \alpha \sin \gamma} \]

This simplifies to:

\[ \cot \beta = \frac{\cos \gamma}{2 \sin \gamma} + \frac{\cos \alpha}{2 \sin \alpha} \]

### Step 5: Relate \(\cot \alpha\) and \(\cot \gamma\)

We know:

\[ \cot \alpha = \frac{\cos \alpha}{\sin \alpha} \]
\[ \cot \gamma = \frac{\cos \gamma}{\sin \gamma} \]

### Step 6: Establish the relationship

Now we can express \(2 \cot \beta\):

\[ 2 \cot \beta = \frac{\cos \gamma}{\sin \gamma} + \frac{\cos \alpha}{\sin \alpha} \]

This can be rewritten as:

\[ 2 \cot \beta = \cot \gamma + \cot \alpha \]

### Conclusion: Determine the relationship

The equation \(2 \cot \beta = \cot \alpha + \cot \gamma\) indicates that \(\cot \alpha\), \(\cot \beta\), and \(\cot \gamma\) are in Arithmetic Progression (AP).

Thus, the final answer is:

**Cotangent values are in AP.**

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