The total number of solution of `|cotx|=cotx+1/(sinx),x in [0,3pi]` , is equal to 1 (b) 2 (c) 3 (d) 0

Options

A. 1
B. 2
C. 3
D. 0

Explanation

To solve the equation \( | \cot x | = \cot x + \frac{1}{\sin x} \) for \( x \) in the interval \( [0, 3\pi] \), we will analyze the cases based on the definition of the absolute value.

### Step 1: Analyze the cases for \( | \cot x | \)

The absolute value function splits the equation into two cases:

1. **Case 1:** \( \cot x \geq 0 \)
\[ | \cot x | = \cot x \]
Therefore, the equation becomes:
\[ \cot x = \cot x + \frac{1}{\sin x} \]
Simplifying this gives:
\[ 0 = \frac{1}{\sin x} \]
This is impossible since \( \frac{1}{\sin x} \) cannot be zero. Thus, there are no solutions in this case.

2. **Case 2:** \( \cot x < 0 \)
\[ | \cot x | = -\cot x \]
Therefore, the equation becomes:
\[ -\cot x = \cot x + \frac{1}{\sin x} \]
Rearranging gives:
\[ -2 \cot x = \frac{1}{\sin x} \]
This can be rewritten as:
\[ \cot x = -\frac{1}{2 \sin x} \]

### Step 2: Use the identity for cotangent

Recall that:
\[ \cot x = \frac{\cos x}{\sin x} \]
Substituting this into our equation gives:
\[ \frac{\cos x}{\sin x} = -\frac{1}{2 \sin x} \]
Multiplying both sides by \( 2 \sin^2 x \) (assuming \( \sin x \neq 0 \)) gives:
\[ 2 \cos x = -1 \]
Thus,
\[ \cos x = -\frac{1}{2} \]

### Step 3: Solve for \( x \)

The solutions for \( \cos x = -\frac{1}{2} \) in the interval \( [0, 3\pi] \) are:
\[ x = \frac{2\pi}{3}, \quad x = \frac{4\pi}{3}, \quad x = \frac{8\pi}{3} \]

### Step 4: Count the solutions

Now we check the values:
- \( x = \frac{2\pi}{3} \) is in the interval.
- \( x = \frac{4\pi}{3} \) is in the interval.
- \( x = \frac{8\pi}{3} \) is also in the interval since \( \frac{8\pi}{3} = 2\pi + \frac{2\pi}{3} \).

Thus, we have three valid solutions.

### Conclusion

The total number of solutions of the equation \( | \cot x | = \cot x + \frac{1}{\sin x} \) in the interval \( [0, 3\pi] \) is **3**.

### Final Answer
(c) 3