The period of `f(x)=cos(abs(sinx)-abs(cosx))` is

Options

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

Answer

To find the period of the function \( f(x) = \cos(|\sin x| - |\cos x|) \), we will analyze the components of the function step by step.

### Step 1: Understand the components of the function
The function consists of the cosine of the expression \( |\sin x| - |\cos x| \). We need to determine the period of this entire expression.

### Step 2: Determine the periods of \( |\sin x| \) and \( |\cos x| \)
The functions \( \sin x \) and \( \cos x \) both have a period of \( 2\pi \). However, since we are taking the absolute values:
- The function \( |\sin x| \) has a period of \( \pi \) because \( |\sin x| = |\sin(x + \pi)| \).
- Similarly, \( |\cos x| \) also has a period of \( \pi \) since \( |\cos x| = |\cos(x + \pi)| \).

### Step 3: Analyze the expression \( |\sin x| - |\cos x| \)
Now, we need to analyze the expression \( |\sin x| - |\cos x| \). Since both \( |\sin x| \) and \( |\cos x| \) have a period of \( \pi \), the expression \( |\sin x| - |\cos x| \) will also have a period of \( \pi \).

### Step 4: Determine the period of \( f(x) \)
Since \( f(x) = \cos(|\sin x| - |\cos x|) \) is the cosine of the expression \( |\sin x| - |\cos x| \), and since \( \cos \) is a periodic function with a period of \( 2\pi \), we need to check if the argument \( |\sin x| - |\cos x| \) has a period that divides \( 2\pi \).

Given that \( |\sin x| - |\cos x| \) has a period of \( \pi \), the function \( f(x) \) will also have a period of \( \pi \).

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

### Final Answer
The period of \( f(x) \) is \( \pi \).

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