Period of the function
`f(x)=sin((x)/(2))cos [(x)/(2)]-cos((x)/(2))sin[(x)/(2)]`, where [.] denotes the greatest integer function, is _________.

Explanation

To find the period of the function \( f(x) = \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right) - \cos\left(\frac{x}{2}\right) \sin\left(\frac{x}{2}\right) \), we can simplify the expression first.

### Step 1: Simplify the function
Using the identity \( \sin A \cos B - \cos A \sin B = \sin(A - B) \), we can rewrite the function:
\[ f(x) = \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right) - \cos\left(\frac{x}{2}\right) \sin\left(\frac{x}{2}\right) = \sin\left(\frac{x}{2} - \frac{x}{2}\right) = \sin(0) = 0 \]

### Step 2: Analyze the function
Since the function simplifies to \( f(x) = 0 \), it is a constant function. The period of a constant function is defined as the smallest positive value \( T \) such that \( f(x + T) = f(x) \) for all \( x \).

### Step 3: Determine the period
For a constant function, \( f(x) \) does not change with \( x \). Therefore, it holds true for any value of \( T \). However, we typically define the period of a constant function as:
\[ \text{Period} = 0 \]

### Final Answer
The period of the function \( f(x) \) is \( 0 \).

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