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 α, β, and γ based on the given equations and then compute the expression \(\frac{|\alpha - \beta|}{\beta + \gamma}\).

### Step 1: Solve for α and β from \( \cot x = -\sqrt{3} \)

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

- In the second quadrant, \( x = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \)
- In the fourth quadrant, \( x = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3} \)

Thus, the solutions for \( \cot x = -\sqrt{3} \) in the interval \([0, 2\pi]\) are:
- \( \alpha = \frac{2\pi}{3} \)
- \( \beta = \frac{5\pi}{3} \)

### Step 2: Solve for α and γ from \( \csc x = -2 \)

The equation \( \csc x = -2 \) implies that \( \sin x = -\frac{1}{2} \). The sine function is negative in the third and fourth quadrants.

- In the third quadrant, \( x = \pi + \frac{\pi}{6} = \frac{7\pi}{6} \)
- In the fourth quadrant, \( x = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6} \)

Thus, the solutions for \( \csc x = -2 \) in the interval \([0, 2\pi]\) are:
- \( \alpha = \frac{11\pi}{6} \)
- \( \gamma = \frac{7\pi}{6} \)

### Step 3: Identify the common root

From the two sets of solutions, we can see that:
- From \( \cot x = -\sqrt{3} \): \( \alpha = \frac{2\pi}{3}, \beta = \frac{5\pi}{3} \)
- From \( \csc x = -2 \): \( \alpha = \frac{11\pi}{6}, \gamma = \frac{7\pi}{6} \)

The common root is \( \alpha = \frac{11\pi}{6} \).

### Step 4: Assign values to β and γ

From the equations, we have:
- \( \beta = \frac{5\pi}{3} \)
- \( \gamma = \frac{7\pi}{6} \)

### Step 5: Calculate \( |\alpha - \beta| \)

Now we compute \( |\alpha - \beta| \):
\[ |\alpha - \beta| = \left| \frac{11\pi}{6} - \frac{5\pi}{3} \right| \]
To subtract these, we convert \( \frac{5\pi}{3} \) to sixths:
\[ \frac{5\pi}{3} = \frac{10\pi}{6} \]
Thus,
\[ |\alpha - \beta| = \left| \frac{11\pi}{6} - \frac{10\pi}{6} \right| = \left| \frac{\pi}{6} \right| = \frac{\pi}{6} \]

### Step 6: Calculate \( \beta + \gamma \)

Next, we compute \( \beta + \gamma \):
\[ \beta + \gamma = \frac{5\pi}{3} + \frac{7\pi}{6} \]
Converting \( \frac{5\pi}{3} \) to sixths:
\[ \frac{5\pi}{3} = \frac{10\pi}{6} \]
Thus,
\[ \beta + \gamma = \frac{10\pi}{6} + \frac{7\pi}{6} = \frac{17\pi}{6} \]

### Step 7: Compute the final expression

Now, we can compute the final expression:
\[ \frac{|\alpha - \beta|}{\beta + \gamma} = \frac{\frac{\pi}{6}}{\frac{17\pi}{6}} = \frac{1}{17} \]

### Final Answer

Thus, the value of \( \frac{|\alpha - \beta|}{\beta + \gamma} \) is \( \frac{1}{17} \).

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