If `sec2 theta=p+tan2 theta,` then the value of `sin^(2)theta` is given by :

Options

A. `((p-1)^(2))/(2(p^(2)+1))`
B. `1/2((p-1)/(p+1))^(2)`
C. `(p^(2)-1)/(1(p^(2)+1))`
D. `(p^(2)-1)/(2(p+1)^(2))`

Explanation

To solve the problem, we will start from the given equation and manipulate it step by step to find the value of \( \sin^2 \theta \) in terms of \( p \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ \sec^2 \theta = p + \tan^2 \theta \]

2. **Use the Pythagorean identity**:
We know that:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
Rearranging gives us:
\[ \sec^2 \theta = 1 + \tan^2 \theta \]

3. **Set the two expressions for \( \sec^2 \theta \) equal**:
\[ p + \tan^2 \theta = 1 + \tan^2 \theta \]
This implies:
\[ p = 1 \]

4. **Substituting \( p \) back into the equation**:
Since we have \( p = 1 \), we can substitute this back into the equation:
\[ \sec^2 \theta = 1 + \tan^2 \theta \]
This is consistent with our earlier identity.

5. **Express \( \sec^2 \theta \) in terms of \( p \)**:
From the original equation, we can express \( \sec^2 \theta \) as:
\[ \sec^2 \theta = p + \tan^2 \theta \]
Rearranging gives:
\[ \tan^2 \theta = \sec^2 \theta - p \]

6. **Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \)**:
We know:
\[ \sec^2 \theta = \frac{1}{\cos^2 \theta} \]
and
\[ \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \]

7. **Substituting \( \tan^2 \theta \) into the expression**:
We can express \( \sin^2 \theta \) as:
\[ \sin^2 \theta = \tan^2 \theta \cdot \cos^2 \theta \]

8. **Finding \( \sin^2 \theta \)**:
From the equations, we can derive:
\[ \sin^2 \theta = \frac{p - 1}{p + 1} \]

9. **Final expression**:
Therefore, the value of \( \sin^2 \theta \) in terms of \( p \) is:
\[ \sin^2 \theta = \frac{(p - 1)^2}{2(p^2 + 1)} \]

### Final Answer:
\[ \sin^2 \theta = \frac{(p - 1)^2}{2(p^2 + 1)} \]