To simplify the given expression:

\[
\frac{1}{\cos^2(A)} \cdot \frac{1}{\sin^2(A)}
\]

we can rewrite it as:

\[
\frac{1}{\cos^2(A) \sin^2(A)}.
\]

Now, we use the identity:

\[
\sin^2(A) \cos^2(A) = \frac{1}{4} \sin^2(2A),
\]

so the expression becomes:

\[
\frac{1}{\frac{1}{4} \sin^2(2A)} = \frac{4}{\sin^2(2A)}.
\]

Thus, the simplified form is:

\[
\frac{4}{\sin^2(2A)}.
\]

Would you like more details on any step? Here are some related questions for further exploration:

1. How can trigonometric identities be used to simplify more complex expressions?
2. What is the derivation of the double-angle identity for sine, \(\sin(2A)\)?
3. How does \(\sin^2(A) + \cos^2(A) = 1\) relate to other trigonometric identities?
4. What are some common mistakes when manipulating trigonometric expressions?
5. Can this expression be interpreted geometrically?

**Tip:** Memorizing common trigonometric identities, such as double-angle and Pythagorean identities, can significantly simplify solving problems.

Explore how to simplify the trigonometric expression 1 ÷ cos²(A) ⋅ 1 ÷ sin²(A) step-by-step using the Pythagorean identity and double-angle formulas. This problem involves key trigonometric identities and is suitable for students in Grades 9-12.

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