The period of function `(|sin x|+|cos x|)/(|sin x-cos x|+|sin x+cos x|)` is :

Options

A. `pi`
B. `pi/2`
C. `2pi`
D. `(2pi)/(3)`

Answer

To find the period of the function \( f(x) = \frac{|\sin x| + |\cos x|}{|\sin x - \cos x| + |\sin x + \cos x|} \), we will analyze both the numerator and the denominator separately.

### Step 1: Analyze the Numerator
The numerator is \( |\sin x| + |\cos x| \).

- The function \( |\sin x| \) has a period of \( \pi \).
- The function \( |\cos x| \) also has a period of \( \pi \).

However, since both functions are combined in a sum, we need to find the least common period. The period of \( |\sin x| + |\cos x| \) is \( \frac{\pi}{2} \).

### Step 2: Analyze the Denominator
The denominator is \( |\sin x - \cos x| + |\sin x + \cos x| \).

1. **For \( |\sin x - \cos x| \)**:
- We can rewrite it as \( |\sqrt{2} \left( \frac{1}{\sqrt{2}} \sin x - \frac{1}{\sqrt{2}} \cos x \right)| \).
- This can be expressed using the sine function:
\[ |\sin x - \cos x| = \sqrt{2} |\sin(x - \frac{\pi}{4})| \]
- The period of \( |\sin(x - \frac{\pi}{4})| \) is \( \pi \).

2. **For \( |\sin x + \cos x| \)**:
- Similarly, we can rewrite it as \( |\sqrt{2} \left( \frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x \right)| \).
- This can also be expressed using the sine function:
\[ |\sin x + \cos x| = \sqrt{2} |\sin(x + \frac{\pi}{4})| \]
- The period of \( |\sin(x + \frac{\pi}{4})| \) is also \( \pi \).

### Step 3: Combine the Denominator
The denominator \( |\sin x - \cos x| + |\sin x + \cos x| \) combines two functions, both having a period of \( \pi \). Therefore, the period of the denominator is also \( \pi \).

### Step 4: Determine the Overall Period
Now we have:
- Period of the numerator: \( \frac{\pi}{2} \)
- Period of the denominator: \( \pi \)

To find the overall period of the function \( f(x) \), we take the least common multiple (LCM) of the two periods:

\[ \text{LCM}\left(\frac{\pi}{2}, \pi\right) = \pi \]

### Conclusion
Thus, the period of the function \( f(x) = \frac{|\sin x| + |\cos x|}{|\sin x - \cos x| + |\sin x + \cos x|} \) is \( \pi \).

### Final Answer
The period of the function is \( \pi \).

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