`f(x)=(cosx)/([(2x)/pi]+1/2),` where `x` is not an integral multiple of `pi` and `[dot]` denotes the greatest integer function, is an odd function an even function neither odd nor even none of these

Options

A. an odd function
B. an even function
C. neither odd nor even
D. None of these

Explanation

To determine whether the function \( f(x) = \frac{\cos x}{\left[\frac{2x}{\pi}\right] + \frac{1}{2}} \) is an odd function, an even function, or neither, we will follow these steps:

### Step 1: Define the function
We start with the given function:
\[ f(x) = \frac{\cos x}{\left[\frac{2x}{\pi}\right] + \frac{1}{2}} \]
where \( [\cdot] \) denotes the greatest integer function.

### Step 2: Find \( f(-x) \)
To check if the function is odd or even, we need to find \( f(-x) \):
\[ f(-x) = \frac{\cos(-x)}{\left[\frac{2(-x)}{\pi}\right] + \frac{1}{2}} \]
Using the property of cosine, \( \cos(-x) = \cos x \), we can rewrite \( f(-x) \) as:
\[ f(-x) = \frac{\cos x}{\left[-\frac{2x}{\pi}\right] + \frac{1}{2}} \]

### Step 3: Simplify the denominator
Next, we need to simplify the denominator:
\[ \left[-\frac{2x}{\pi}\right] + \frac{1}{2} \]
The greatest integer function \( \left[-\frac{2x}{\pi}\right] \) can be expressed as:
\[ \left[-\frac{2x}{\pi}\right] = -1 - \left[\frac{2x}{\pi}\right] \quad \text{(since } x \text{ is not an integral multiple of } \pi\text{)} \]
Thus, we have:
\[ f(-x) = \frac{\cos x}{-1 - \left[\frac{2x}{\pi}\right] + \frac{1}{2}} = \frac{\cos x}{-\left[\frac{2x}{\pi}\right] - \frac{1}{2}} \]

### Step 4: Compare \( f(-x) \) with \( -f(x) \)
Now, we need to analyze \( -f(x) \):
\[ -f(x) = -\frac{\cos x}{\left[\frac{2x}{\pi}\right] + \frac{1}{2}} = \frac{-\cos x}{\left[\frac{2x}{\pi}\right] + \frac{1}{2}} \]
We want to see if \( f(-x) = -f(x) \):
\[ f(-x) = \frac{\cos x}{-\left[\frac{2x}{\pi}\right] - \frac{1}{2}} = \frac{-\cos x}{\left[\frac{2x}{\pi}\right] + \frac{1}{2}} \]
This shows that:
\[ f(-x) = -f(x) \]
Thus, \( f(x) \) is an odd function.

### Conclusion
The function \( f(x) \) is an odd function.

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