The peroid of the function `f(x) =(|sinx|-|cosx|)/(|sin x + cosx|)` is

Options

A. `(pi)/(2)`
B. `2pi`
C. `pi `
D. none of these

Explanation

To find the period of the function \( f(x) = \frac{|\sin x| - |\cos x|}{|\sin x + \cos x|} \), we will analyze the periodicity of the components involved in the function.

### Step 1: Understanding the Components
The function consists of the absolute values of sine and cosine functions. Both \( |\sin x| \) and \( |\cos x| \) have a period of \( \pi \), because they repeat their values every \( \pi \) radians.

**Hint:** Check the periodicity of each component of the function separately.

### Step 2: Check the Period of the Numerator
The numerator is \( |\sin x| - |\cos x| \). Since both \( |\sin x| \) and \( |\cos x| \) have a period of \( \pi \), the expression \( |\sin x| - |\cos x| \) also has a period of \( \pi \).

**Hint:** Verify if the subtraction of two periodic functions with the same period retains that period.

### Step 3: Check the Period of the Denominator
The denominator is \( |\sin x + \cos x| \). To find its period, we can analyze the expression:

\[ |\sin x + \cos x| = |\sqrt{2} \sin\left(x + \frac{\pi}{4}\right)| \]

This transformation shows that \( |\sin x + \cos x| \) also has a period of \( 2\pi \) since the sine function has a period of \( 2\pi \).

**Hint:** Use trigonometric identities to simplify the expression and find its period.

### Step 4: Determine the Overall Period
Since the numerator has a period of \( \pi \) and the denominator has a period of \( 2\pi \), we need to find the least common multiple (LCM) of these two periods to determine the overall period of the function.

The LCM of \( \pi \) and \( 2\pi \) is \( 2\pi \).

**Hint:** Remember that the overall period is determined by the longest period among the components.

### Step 5: Check for Fundamental Period
Next, we check if the function is periodic with \( \pi \). We can evaluate \( f(x + \pi) \):

\[ f(x + \pi) = \frac{|\sin(x + \pi)| - |\cos(x + \pi)|}{|\sin(x + \pi) + \cos(x + \pi)|} \]
Using the properties of sine and cosine:
\[ = \frac{|- \sin x| - |-\cos x|}{|-\sin x - \cos x|} = \frac{|\sin x| - |\cos x|}{|-\sin x - \cos x|} = f(x) \]

This shows that \( f(x) \) is periodic with period \( \pi \).

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

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

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