Three positive acute angles `alpha, beta and gamma` satisfy the relation `tan. (beta)/(2)=(1)/(3)cot.(alpha)/(2)and cot.(gamma)/(2)=(1)/(2)(3tan.(alpha)/(2)+cot.(alpha)/(2))`. Then, the value of `alpha+beta+gamma` is equal to

Options

A. `pi`
B. `2pi`
C. `(pi)/(2)`
D. `(3pi)/(2)`

Answer

To solve the problem, we need to analyze the given equations step by step.

### Step 1: Understand the given equations
We are given two relations involving the angles \( \alpha, \beta, \) and \( \gamma \):
1. \(\tan\left(\frac{\beta}{2}\right) = \frac{1}{3} \cot\left(\frac{\alpha}{2}\right)\)
2. \(\cot\left(\frac{\gamma}{2}\right) = \frac{1}{2}\left(3\tan\left(\frac{\alpha}{2}\right) + \cot\left(\frac{\alpha}{2}\right)\right)\)

### Step 2: Rewrite cotangent in terms of tangent
Using the identity \( \cot(x) = \frac{1}{\tan(x)} \), we can rewrite the first equation:
\[ \tan\left(\frac{\beta}{2}\right) = \frac{1}{3} \cdot \frac{1}{\tan\left(\frac{\alpha}{2}\right)} = \frac{1}{3\tan\left(\frac{\alpha}{2}\right)} \]

### Step 3: Substitute into the first equation
Let \( x = \tan\left(\frac{\alpha}{2}\right) \) and \( y = \tan\left(\frac{\beta}{2}\right) \):
\[ y = \frac{1}{3x} \]

### Step 4: Substitute into the second equation
Now, substitute \( y \) into the second equation:
\[ \cot\left(\frac{\gamma}{2}\right) = \frac{1}{2}\left(3x + \frac{1}{x}\right) \]
This can be rewritten as:
\[ \cot\left(\frac{\gamma}{2}\right) = \frac{3x^2 + 1}{2x} \]

### Step 5: Use the identity for tangent addition
Using the tangent addition formula:
\[ \tan\left(\frac{\alpha + \beta}{2}\right) = \frac{\tan\left(\frac{\alpha}{2}\right) + \tan\left(\frac{\beta}{2}\right)}{1 - \tan\left(\frac{\alpha}{2}\right) \tan\left(\frac{\beta}{2}\right)} \]
Substituting \( y \):
\[ \tan\left(\frac{\alpha + \beta}{2}\right) = \frac{x + \frac{1}{3x}}{1 - x \cdot \frac{1}{3x}} = \frac{x + \frac{1}{3x}}{1 - \frac{1}{3}} = \frac{x + \frac{1}{3x}}{\frac{2}{3}} = \frac{3}{2}\left(x + \frac{1}{3x}\right) \]

### Step 6: Relate to \( \cot\left(\frac{\gamma}{2}\right) \)
From the previous steps, we know:
\[ \tan\left(\frac{\alpha + \beta}{2}\right) = \cot\left(\frac{\gamma}{2}\right) \]
This implies:
\[ \tan\left(\frac{\alpha + \beta}{2}\right) = \frac{2x}{3} \]

### Step 7: Set up the final equation
We now have:
\[ \tan\left(\frac{\alpha + \beta + \gamma}{2}\right) = \infty \]
This implies:
\[ \frac{\alpha + \beta + \gamma}{2} = \frac{\pi}{2} \]
Thus, multiplying by 2 gives:
\[ \alpha + \beta + \gamma = \pi \]

### Final Answer
The value of \( \alpha + \beta + \gamma \) is \( \pi \).

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