What is / are the solutions of the trigonometric equation `"cosec "x+cotx=sqrt(3),"where "0ltxlt2pi`?

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
The cosecant and cotangent functions can be expressed in terms of sine and cosine:
\[ \csc x = \frac{1}{\sin x}, \quad \cot x = \frac{\cos x}{\sin x} \]
Substituting these into the equation gives:
\[ \frac{1}{\sin x} + \frac{\cos x}{\sin x} = \sqrt{3} \]
Combining the fractions, we have:
\[ \frac{1 + \cos x}{\sin x} = \sqrt{3} \]

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

### Step 3: Use the identity for \( \sin x \)
We can express \( \sin x \) in terms of \( \cos x \):
\[ \sin x = \sqrt{1 - \cos^2 x} \]
Substituting this into the equation gives:
\[ 1 + \cos x = \sqrt{3} \sqrt{1 - \cos^2 x} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides results in:
\[ (1 + \cos x)^2 = 3(1 - \cos^2 x) \]
Expanding both sides:
\[ 1 + 2\cos x + \cos^2 x = 3 - 3\cos^2 x \]
Rearranging gives:
\[ 4\cos^2 x + 2\cos x - 2 = 0 \]

### Step 5: Simplify the quadratic equation
Dividing the entire equation by 2:
\[ 2\cos^2 x + \cos x - 1 = 0 \]

### Step 6: Solve the quadratic equation using the quadratic formula
Using the quadratic formula \( \cos x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 2, b = 1, c = -1 \):
\[ \cos x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} \]
\[ \cos x = \frac{-1 \pm \sqrt{1 + 8}}{4} = \frac{-1 \pm 3}{4} \]
Calculating the two possible values:
1. \( \cos x = \frac{2}{4} = \frac{1}{2} \)
2. \( \cos x = \frac{-4}{4} = -1 \)

### Step 7: Find the angles corresponding to \( \cos x \)
1. For \( \cos x = \frac{1}{2} \):
- \( x = \frac{\pi}{3} \) (in the first quadrant)
- \( x = \frac{5\pi}{3} \) (in the fourth quadrant)

2. For \( \cos x = -1 \):
- \( x = \pi \)

### Step 8: List all solutions in the given interval
The solutions in the interval \( 0 < x < 2\pi \) are:
- \( x = \frac{\pi}{3} \)
- \( x = \pi \)
- \( x = \frac{5\pi}{3} \)

### Step 9: Verify the solutions
We need to check if these values satisfy the original equation:
- For \( x = \frac{\pi}{3} \):
\[ \csc\left(\frac{\pi}{3}\right) + \cot\left(\frac{\pi}{3}\right) = 2 + \frac{1}{\sqrt{3}} \quad \text{(valid)} \]
- For \( x = \pi \):
\[ \csc(\pi) + \cot(\pi) \quad \text{(undefined, not valid)} \]
- For \( x = \frac{5\pi}{3} \):
\[ \csc\left(\frac{5\pi}{3}\right) + \cot\left(\frac{5\pi}{3}\right) = -2 + \frac{1}{\sqrt{3}} \quad \text{(valid)} \]

### Final Answer
The valid solutions for the equation \( \csc x + \cot x = \sqrt{3} \) in the interval \( 0 < x < 2\pi \) are:
- \( x = \frac{\pi}{3} \) and \( x = \frac{5\pi}{3} \).