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 \( |\sin x| + |\cos x| \).

1. The functions \( |\sin x| \) and \( |\cos x| \) both have a fundamental period of \( \pi \).
2. To find the period of \( |\sin x| + |\cos x| \), we can check if \( f(x + \pi) = f(x) \):
\[ f(x + \pi) = |\sin(x + \pi)| + |\cos(x + \pi)| \]
Using the properties of sine and cosine:
\[ = |\sin x| + |-\cos x| = |\sin x| + |\cos x| \]
Thus, the period of the numerator \( |\sin x| + |\cos x| \) is \( \pi \).

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

1. Let's denote \( g(x) = |\sin x - \cos x| + |\sin x + \cos x| \).
2. We will check if \( g(x + \pi) = g(x) \):
\[ g(x + \pi) = |\sin(x + \pi) - \cos(x + \pi)| + |\sin(x + \pi) + \cos(x + \pi)| \]
Using the properties of sine and cosine:
\[ = |-\sin x - (-\cos x)| + |-\sin x + (-\cos x)| \]
Simplifying this gives:
\[ = |-\sin x + \cos x| + |-\sin x - \cos x| = |\cos x - \sin x| + |-\sin x - \cos x| \]
Since \( |a| = |-a| \), we can write:
\[ = |\sin x - \cos x| + |\sin x + \cos x| = g(x) \]
Thus, the period of the denominator \( |\sin x - \cos x| + |\sin x + \cos x| \) is also \( \pi \).

### Step 3: Determine the Overall Period
Since both the numerator and denominator have a period of \( \pi \), we can conclude that the overall period of the function \( f(x) \) is also \( \pi \).

### Final Answer
The period of the function \( \frac{|\sin x| + |\cos x|}{|\sin x - \cos x| + |\sin x + \cos x|} \) is \( \pi \).