If `alpha and beta` are the solution of `cotx=-sqrt3` in `[0, 2pi]` and `alpha and gamma` are the roots of `"cosec x"=-2` in `[0, 2pi]`, then the value of `(|alpha-beta|)/(beta+gamma)` is equal to

Options

A. `(1)/(2)`
B. `2`
C. `(1)/(3)`
D. `3`

Answer

To solve the problem, we need to find the values of \( \alpha \), \( \beta \), and \( \gamma \) based on the given equations and then compute the expression \( \frac{|\alpha - \beta|}{\beta + \gamma} \).

### Step 1: Find \( \alpha \) and \( \beta \) from \( \cot x = -\sqrt{3} \)

The equation \( \cot x = -\sqrt{3} \) indicates that \( x \) is in the second and fourth quadrants since cotangent is negative in these quadrants.

We know that:
\[ \cot \frac{\pi}{6} = \sqrt{3} \]

Thus, for \( \cot x = -\sqrt{3} \):
- In the second quadrant:
\[ x = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \]
- In the fourth quadrant:
\[ x = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6} \]

So, we have:
\[ \alpha = \frac{5\pi}{6}, \quad \beta = \frac{11\pi}{6} \]

### Step 2: Find \( \alpha \) and \( \gamma \) from \( \csc x = -2 \)

The equation \( \csc x = -2 \) indicates that \( x \) is also in the third and fourth quadrants since cosecant is negative in these quadrants.

We know that:
\[ \csc \frac{\pi}{6} = 2 \]

Thus, for \( \csc x = -2 \):
- In the third quadrant:
\[ x = \pi + \frac{\pi}{6} = \frac{7\pi}{6} \]
- In the fourth quadrant:
\[ x = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6} \]

So, we have:
\[ \alpha = \frac{11\pi}{6}, \quad \gamma = \frac{7\pi}{6} \]

### Step 3: Identify the values of \( \alpha \), \( \beta \), and \( \gamma \)

From the above steps, we have:
- \( \alpha = \frac{11\pi}{6} \)
- \( \beta = \frac{5\pi}{6} \)
- \( \gamma = \frac{7\pi}{6} \)

### Step 4: Calculate \( |\alpha - \beta| \)

Now we compute \( |\alpha - \beta| \):
\[ |\alpha - \beta| = \left| \frac{11\pi}{6} - \frac{5\pi}{6} \right| = \left| \frac{6\pi}{6} \right| = \pi \]

### Step 5: Calculate \( \beta + \gamma \)

Next, we compute \( \beta + \gamma \):
\[ \beta + \gamma = \frac{5\pi}{6} + \frac{7\pi}{6} = \frac{12\pi}{6} = 2\pi \]

### Step 6: Compute the final expression

Now we can compute the desired expression:
\[ \frac{|\alpha - \beta|}{\beta + \gamma} = \frac{\pi}{2\pi} = \frac{1}{2} \]

### Final Answer
Thus, the value of \( \frac{|\alpha - \beta|}{\beta + \gamma} \) is \( \frac{1}{2} \).