If `x in [0,2pi]" then "y_(1)=(sin x)/(|sin x|), y_(2)=(|cos x|)/(cos x)` are identical functions for ` x in :`
I. `(0,pi/2)" "II. (pi/2, pi)," "III. (pi,(3pi)/(2))," "IV. ((3pi)/(2),2pi)`

Options

A. I,II
B. I,III
C. II,III
D. I,IV

Answer

To determine where the functions \( y_1 = \frac{\sin x}{|\sin x|} \) and \( y_2 = \frac{|\cos x|}{\cos x} \) are identical, we need to analyze both functions over the intervals provided.

### Step 1: Analyze \( y_1 = \frac{\sin x}{|\sin x|} \)

- **For \( x \in (0, \frac{\pi}{2}) \)**:
- \( \sin x > 0 \)
- Therefore, \( y_1 = \frac{\sin x}{\sin x} = 1 \)

- **For \( x \in (\frac{\pi}{2}, \pi) \)**:
- \( \sin x > 0 \)
- Therefore, \( y_1 = \frac{\sin x}{\sin x} = 1 \)

- **For \( x \in (\pi, \frac{3\pi}{2}) \)**:
- \( \sin x < 0 \)
- Therefore, \( y_1 = \frac{\sin x}{|\sin x|} = -1 \)

- **For \( x \in (\frac{3\pi}{2}, 2\pi) \)**:
- \( \sin x < 0 \)
- Therefore, \( y_1 = \frac{\sin x}{|\sin x|} = -1 \)

### Step 2: Analyze \( y_2 = \frac{|\cos x|}{\cos x} \)

- **For \( x \in (0, \frac{\pi}{2}) \)**:
- \( \cos x > 0 \)
- Therefore, \( y_2 = \frac{\cos x}{\cos x} = 1 \)

- **For \( x \in (\frac{\pi}{2}, \pi) \)**:
- \( \cos x < 0 \)
- Therefore, \( y_2 = \frac{-\cos x}{\cos x} = -1 \)

- **For \( x \in (\pi, \frac{3\pi}{2}) \)**:
- \( \cos x < 0 \)
- Therefore, \( y_2 = \frac{-\cos x}{\cos x} = -1 \)

- **For \( x \in (\frac{3\pi}{2}, 2\pi) \)**:
- \( \cos x > 0 \)
- Therefore, \( y_2 = \frac{\cos x}{\cos x} = 1 \)

### Step 3: Compare the Functions

Now we compare \( y_1 \) and \( y_2 \) in each interval:

1. **Interval \( (0, \frac{\pi}{2}) \)**:
- \( y_1 = 1 \)
- \( y_2 = 1 \)
- **Identical**.

2. **Interval \( (\frac{\pi}{2}, \pi) \)**:
- \( y_1 = 1 \)
- \( y_2 = -1 \)
- **Not identical**.

3. **Interval \( (\pi, \frac{3\pi}{2}) \)**:
- \( y_1 = -1 \)
- \( y_2 = -1 \)
- **Identical**.

4. **Interval \( (\frac{3\pi}{2}, 2\pi) \)**:
- \( y_1 = -1 \)
- \( y_2 = 1 \)
- **Not identical**.

### Conclusion

The functions \( y_1 \) and \( y_2 \) are identical in the intervals:
- I. \( (0, \frac{\pi}{2}) \)
- III. \( (\pi, \frac{3\pi}{2}) \)

Thus, the correct options are I and III.