The number of values of `x int [-2pi, 2pi]` satisfying `tanx + cotx = 2" cosec x"` is

Options

A. 2
B. 4
C. 6
D. 8

Answer

To solve the equation \( \tan x + \cot x = 2 \csc x \) for \( x \) in the interval \( [-2\pi, 2\pi] \), we will follow these steps:

### Step 1: Rewrite the equation using sine and cosine
We know that:
\[ \tan x = \frac{\sin x}{\cos x}, \quad \cot x = \frac{\cos x}{\sin x}, \quad \text{and} \quad \csc x = \frac{1}{\sin x} \]
Substituting these into the equation gives:
\[ \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = 2 \cdot \frac{1}{\sin x} \]

### Step 2: Find a common denominator
The common denominator for the left-hand side is \( \sin x \cos x \):
\[ \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} = \frac{2}{\sin x} \]

### Step 3: Simplify using the Pythagorean identity
Using the identity \( \sin^2 x + \cos^2 x = 1 \), we can simplify the equation:
\[ \frac{1}{\sin x \cos x} = \frac{2}{\sin x} \]

### Step 4: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ 1 = 2 \cos x \]

### Step 5: Solve for \( \cos x \)
Rearranging the equation, we find:
\[ \cos x = \frac{1}{2} \]

### Step 6: Find the general solutions for \( \cos x = \frac{1}{2} \)
The values of \( x \) where \( \cos x = \frac{1}{2} \) are:
\[ x = \frac{\pi}{3} + 2k\pi \quad \text{and} \quad x = -\frac{\pi}{3} + 2k\pi \quad \text{for integers } k \]

### Step 7: Determine the specific solutions in the interval \( [-2\pi, 2\pi] \)
For \( k = -2, -1, 0, 1, 2 \):
- For \( k = -2 \):
- \( x = \frac{\pi}{3} - 4\pi = -\frac{11\pi}{3} \) (not in range)
- \( x = -\frac{\pi}{3} - 4\pi = -\frac{13\pi}{3} \) (not in range)

- For \( k = -1 \):
- \( x = \frac{\pi}{3} - 2\pi = -\frac{5\pi}{3} \)
- \( x = -\frac{\pi}{3} - 2\pi = -\frac{7\pi}{3} \) (not in range)

- For \( k = 0 \):
- \( x = \frac{\pi}{3} \)
- \( x = -\frac{\pi}{3} \)

- For \( k = 1 \):
- \( x = \frac{\pi}{3} + 2\pi = \frac{7\pi}{3} \) (not in range)
- \( x = -\frac{\pi}{3} + 2\pi = \frac{5\pi}{3} \)

- For \( k = 2 \):
- \( x = \frac{\pi}{3} + 4\pi = \frac{13\pi}{3} \) (not in range)
- \( x = -\frac{\pi}{3} + 4\pi = \frac{11\pi}{3} \) (not in range)

### Step 8: Collect the valid solutions
The valid solutions in the interval \( [-2\pi, 2\pi] \) are:
1. \( \frac{\pi}{3} \)
2. \( -\frac{\pi}{3} \)
3. \( -\frac{5\pi}{3} \)
4. \( \frac{5\pi}{3} \)

### Conclusion
Thus, the total number of solutions is **4**.