Find the period of the functions (where [ * ] denotes greatest integer function)
`f(x) = (1)/( 1 - cos x )`

Answer

To find the period of the function \( f(x) = \frac{1}{1 - \cos x} \), we can follow these steps:

### Step 1: Identify the periodic component
The function \( f(x) \) is defined in terms of \( \cos x \). The cosine function \( \cos x \) is periodic with a period of \( 2\pi \).

### Step 2: Analyze the effect of the transformation
The function \( f(x) \) involves a transformation of \( \cos x \) through the expression \( 1 - \cos x \). Since the transformation involves addition and subtraction of constants, it does not affect the periodicity of the cosine function.

### Step 3: Determine the period of \( f(x) \)
Since \( \cos x \) has a period of \( 2\pi \), and the transformation \( 1 - \cos x \) does not change this period, the function \( f(x) = \frac{1}{1 - \cos x} \) will also have the same period as \( \cos x \).

### Step 4: Conclusion
Thus, the period of the function \( f(x) = \frac{1}{1 - \cos x} \) is \( 2\pi \).

### Final Answer
The period of the function \( f(x) = \frac{1}{1 - \cos x} \) is \( 2\pi \).

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