Find the value of `(1-sin^(2)A)(1+tan^(2)A)`.

Answer

To find the value of \( (1 - \sin^2 A)(1 + \tan^2 A) \), we can follow these steps:

### Step 1: Rewrite the expression
The given expression is:
\[ (1 - \sin^2 A)(1 + \tan^2 A) \]

### Step 2: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ 1 - \sin^2 A = \cos^2 A \]
So we can substitute this into our expression:
\[ \cos^2 A (1 + \tan^2 A) \]

### Step 3: Use the identity for \( \tan^2 A \)
Recall that:
\[ 1 + \tan^2 A = \sec^2 A \]
Thus, we can rewrite the expression as:
\[ \cos^2 A \cdot \sec^2 A \]

### Step 4: Simplify the expression
Since \( \sec A = \frac{1}{\cos A} \), we have:
\[ \sec^2 A = \frac{1}{\cos^2 A} \]
Therefore, substituting this in gives:
\[ \cos^2 A \cdot \frac{1}{\cos^2 A} \]

### Step 5: Cancel the terms
The \( \cos^2 A \) in the numerator and denominator cancels out:
\[ 1 \]

### Conclusion
Thus, the value of \( (1 - \sin^2 A)(1 + \tan^2 A) \) is:
\[ \boxed{1} \]

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