The period of the function
`f(x)=|sinx|-|cosx|` , is

Options

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

Explanation

To find the period of the function \( f(x) = |\sin x| - |\cos x| \), we will analyze the periodicity of each component of the function and then determine the overall period.

### Step 1: Identify the periods of the individual components
The function consists of two parts: \( |\sin x| \) and \( |\cos x| \).

- The period of \( \sin x \) is \( 2\pi \). When we take the absolute value, the period of \( |\sin x| \) becomes \( \pi \) because \( |\sin x| \) repeats its values every \( \pi \) radians.

- Similarly, the period of \( \cos x \) is also \( 2\pi \). Therefore, the period of \( |\cos x| \) is also \( \pi \).

### Step 2: Determine the overall period
Since both \( |\sin x| \) and \( |\cos x| \) have the same period of \( \pi \), we can find the overall period of the function \( f(x) \) by taking the least common multiple (LCM) of the periods of the individual components.

\[ \text{LCM}(\pi, \pi) = \pi \]

### Step 3: Verify if the function is periodic with period \( \pi \)
To confirm that \( f(x) \) is periodic with period \( \pi \), we need to check if:

\[ f(x + \pi) = f(x) \]

Calculating \( f(x + \pi) \):

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

Since \( f(x + \pi) = f(x) \), we confirm that the function is periodic with period \( \pi \).

### Conclusion
Thus, the period of the function \( f(x) = |\sin x| - |\cos x| \) is:

\[ \boxed{\pi} \]