Statement-1: The period of the function `f(x)=cos[2pi]^(2)x+cos[-2pi^(2)]x+[x]` is ` pi, [x]` being greatest integer function and [x] is a fractional part of x, is `pi` .
Statement-2: The cosine function is periodic with period ` 2pi`

Options

A. Statement-1 is True, Statement-2 is True, statement-2 is a correct explanation for the statement-1 .
B. Statement-1 is True, Statement-2 is True, statement-2 is not a correct explanation for the statement-1 .
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False , Statement-2 is True.

Explanation

To solve the problem, we need to analyze the function given in Statement-1 and determine its period. The function is:

\[ f(x) = \cos(2\pi^2 x) + \cos(-2\pi^2 x) + [x] \]

where \([x]\) is the greatest integer function (also known as the floor function).

### Step 1: Analyze the cosine terms
The cosine function is periodic with a period of \(2\pi\). However, we need to determine the period of the specific terms in our function.

1. **For the term \(\cos(2\pi^2 x)\)**:
- The period \(T_1\) can be found using the formula for the period of cosine:
\[ T_1 = \frac{2\pi}{k} \quad \text{where } k \text{ is the coefficient of } x. \]
Here, \(k = 2\pi^2\), so:
\[ T_1 = \frac{2\pi}{2\pi^2} = \frac{1}{\pi}. \]

2. **For the term \(\cos(-2\pi^2 x)\)**:
- The period \(T_2\) is the same as \(T_1\) because cosine is an even function:
\[ T_2 = \frac{2\pi}{-2\pi^2} = \frac{1}{\pi}. \]

### Step 2: Analyze the greatest integer function
The term \([x]\) is the greatest integer function, which is not periodic. It takes integer values and jumps at every integer point.

### Step 3: Determine the overall period of \(f(x)\)
Since \([x]\) is not periodic, the overall function \(f(x)\) will not have a period. The periodic components (the cosine terms) have a period of \(\frac{1}{\pi}\), but the presence of the non-periodic term \([x]\) means that the entire function does not repeat itself over any interval.

### Conclusion for Statement-1
Thus, the period of the function \(f(x)\) is not \(\pi\) as stated. Therefore, Statement-1 is **false**.

### Step 4: Analyze Statement-2
The cosine function is indeed periodic with a period of \(2\pi\). This is a well-known property of the cosine function. Therefore, Statement-2 is **true**.

### Final Answer
- Statement-1: False
- Statement-2: True