What is/are the solutions of the trigonometric equation `cosec x + cotx = sqrt3`. where `0 lt x lt 2x`?

Options

A. `(5pi)/(3)` only
B. `(pi)/(3)` only
C. `pi` only
D. `pi,(pi)/(3),(5pi)/(3)`

Answer

To solve the trigonometric equation \( \csc x + \cot x = \sqrt{3} \) for \( 0 < x < 2\pi \), we can follow these steps:

### Step 1: Rewrite the equation in terms of sine and cosine
We know that:
\[ \csc x = \frac{1}{\sin x} \quad \text{and} \quad \cot x = \frac{\cos x}{\sin x} \]
Thus, we can rewrite the equation as:
\[ \frac{1}{\sin x} + \frac{\cos x}{\sin x} = \sqrt{3} \]
Combining the fractions gives:
\[ \frac{1 + \cos x}{\sin x} = \sqrt{3} \]

### Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying yields:
\[ 1 + \cos x = \sqrt{3} \sin x \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ \sqrt{3} \sin x - \cos x - 1 = 0 \]

### Step 4: Use the identity for sine and cosine
We can express this in a more manageable form. We know that \( \sin x = \sqrt{1 - \cos^2 x} \). However, in this case, it might be easier to isolate \( \sin x \):
\[ \sin x = \frac{1 + \cos x}{\sqrt{3}} \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides gives:
\[ \sin^2 x = \frac{(1 + \cos x)^2}{3} \]
Using the identity \( \sin^2 x + \cos^2 x = 1 \), we substitute \( \sin^2 x \):
\[ 1 - \cos^2 x = \frac{(1 + \cos x)^2}{3} \]

### Step 6: Expand and simplify
Expanding the right side:
\[ 1 - \cos^2 x = \frac{1 + 2\cos x + \cos^2 x}{3} \]
Multiplying through by 3 to eliminate the fraction:
\[ 3 - 3\cos^2 x = 1 + 2\cos x + \cos^2 x \]
Rearranging gives:
\[ 4\cos^2 x + 2\cos x - 2 = 0 \]

### Step 7: Solve the quadratic equation
This can be simplified to:
\[ 2\cos^2 x + \cos x - 1 = 0 \]
Using the quadratic formula \( \cos x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ \cos x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} \]
\[ \cos x = \frac{-1 \pm \sqrt{1 + 8}}{4} = \frac{-1 \pm 3}{4} \]
Calculating the two possible values:
1. \( \cos x = \frac{2}{4} = \frac{1}{2} \)
2. \( \cos x = \frac{-4}{4} = -1 \)

### Step 8: Find the angles corresponding to the cosine values
1. For \( \cos x = \frac{1}{2} \):
- \( x = \frac{\pi}{3} \) and \( x = \frac{5\pi}{3} \)
2. For \( \cos x = -1 \):
- \( x = \pi \)

### Step 9: Compile the solutions
Thus, the solutions for \( 0 < x < 2\pi \) are:
\[ x = \frac{\pi}{3}, \pi, \frac{5\pi}{3} \]

### Final Answer
The solutions of the equation \( \csc x + \cot x = \sqrt{3} \) in the interval \( 0 < x < 2\pi \) are:
\[ x = \frac{\pi}{3}, \pi, \frac{5\pi}{3} \]