Given that `a_(n)=int_(0)^(pi) (sin^(2){(n+1)x})/(sin2x)dx`
Consider the following statements:
1. The sequence `{a_(2n)}` is in AP with common difference zero.
2. The sequence `{a_(2n+1)}` is in AP with common difference zero.
Which of the above statements is/are correct?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Explanation

To solve the problem, we need to analyze the sequence given by the integral

\[ a_n = \int_0^{\pi} \frac{\sin^2((n+1)x)}{\sin(2x)} \, dx. \]

### Step 1: Evaluate the integral \( a_n \)

The integral can be evaluated using properties of definite integrals and trigonometric identities. However, we can also observe that the integral is a definite value that does not depend on \( n \) in terms of its functional form.

### Step 2: Determine the value of \( a_n \)

From the properties of the integral, we can conclude that \( a_n \) is a constant value \( c \) for all \( n \). This means:

\[ a_0 = c, \quad a_1 = c, \quad a_2 = c, \quad \ldots \]

### Step 3: Analyze the sequences \( \{a_{2n}\} \) and \( \{a_{2n+1}\} \)

1. **For the sequence \( \{a_{2n}\} \)**:
- The terms are \( a_0, a_2, a_4, \ldots \).
- Since all these terms are equal to \( c \), we have:
\[ a_{2n} = c \quad \text{for all } n. \]
- The common difference is:
\[ a_{2n+2} - a_{2n} = c - c = 0. \]
- Therefore, the sequence \( \{a_{2n}\} \) is in arithmetic progression (AP) with a common difference of 0.

2. **For the sequence \( \{a_{2n+1}\} \)**:
- The terms are \( a_1, a_3, a_5, \ldots \).
- Similarly, since all these terms are also equal to \( c \), we have:
\[ a_{2n+1} = c \quad \text{for all } n. \]
- The common difference is:
\[ a_{2n+3} - a_{2n+1} = c - c = 0. \]
- Therefore, the sequence \( \{a_{2n+1}\} \) is also in arithmetic progression (AP) with a common difference of 0.

### Conclusion

Both statements are correct:
1. The sequence \( \{a_{2n}\} \) is in AP with a common difference of zero.
2. The sequence \( \{a_{2n+1}\} \) is in AP with a common difference of zero.

Thus, both statements are true.

### Final Answer

Both statements are correct.

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