Choose the correct optioin . Justify your choice.
`(1+ tan^2A)/(1+cot^2A)`=

Options

A. `sec^(2)A`
B. `-1`
C. `cot^(2)A`
D. `tan^(2)A`

Answer

To solve the expression \((1 + \tan^2 A) / (1 + \cot^2 A)\), we can use the trigonometric identities for \(\tan\) and \(\cot\).

### Step-by-Step Solution:

1. **Identify Trigonometric Identities**:
We know the following identities:
- \(\tan^2 A = \frac{\sin^2 A}{\cos^2 A}\)
- \(\cot^2 A = \frac{\cos^2 A}{\sin^2 A}\)
- \(1 + \tan^2 A = \sec^2 A\)
- \(1 + \cot^2 A = \csc^2 A\)

2. **Rewrite the Expression**:
Using the identities, we can rewrite the expression:
\[ \frac{1 + \tan^2 A}{1 + \cot^2 A} = \frac{\sec^2 A}{\csc^2 A} \]

3. **Substituting the Identities**:
We know that:
- \(\sec^2 A = \frac{1}{\cos^2 A}\)
- \(\csc^2 A = \frac{1}{\sin^2 A}\)

Therefore, we can substitute:
\[ \frac{\sec^2 A}{\csc^2 A} = \frac{\frac{1}{\cos^2 A}}{\frac{1}{\sin^2 A}} = \frac{\sin^2 A}{\cos^2 A} \]

4. **Simplifying the Expression**:
The expression \(\frac{\sin^2 A}{\cos^2 A}\) simplifies to:
\[ \tan^2 A \]

5. **Conclusion**:
Thus, we find that:
\[ \frac{1 + \tan^2 A}{1 + \cot^2 A} = \tan^2 A \]

### Final Answer:
The correct option is \(\tan^2 A\).