(1 - cos²A) sec²A is equal to:

The correct answer is tan²A.

We need to simplify the expression: (1 - cos²A) · sec²A

Step 1: Simplify (1 - cos²A) using a Pythagorean Identity.

Recall the fundamental Pythagorean identity:

{\sin }^{2}A+{\cos }^{2}A=1

Rearranging this gives us:

1-{\cos }^{2}A={\sin }^{2}A

So the expression becomes:

{\sin }^{2}A·{\sec }^{2}A

Step 2: Replace sec²A with its definition.

By definition, {\sec }^{2}A=\frac{1}{{\cos }^{2}A}

Substituting this in:

{\sin }^{2}A·\frac{1}{{\cos }^{2}A}

Step 3: Combine and simplify.

This gives us:

\frac{{\sin }^{2}A}{{\cos }^{2}A}

Step 4: Recognize the result as a known identity.

Since \tan A=\frac{\sin A}{\cos A}, it follows that:

\frac{{\sin }^{2}A}{{\cos }^{2}A}={\tan }^{2}A

Conclusion:

(1 - cos²A) · sec²A = sin²A · sec²A = \frac{{\sin }^{2}A}{{\cos }^{2}A} = tan²A