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Let's solve the given trigonometric identity:

\[
\frac{\sin^4 A + \cos^4 A}{1 - 2 \sin^2 A \cos^2 A} = 1
\]

### Step 1: Simplify the numerator
We can simplify the expression \(\sin^4 A + \cos^4 A\) using the identity:

\[
\sin^4 A + \cos^4 A = (\sin^2 A + \cos^2 A)^2 - 2 \sin^2 A \cos^2 A
\]

Since \(\sin^2 A + \cos^2 A = 1\), the expression simplifies to:

\[
\sin^4 A + \cos^4 A = 1 - 2 \sin^2 A \cos^2 A
\]

### Step 2: Substitute the simplified expression into the original equation
Now, substitute this result back into the original equation:

\[
\frac{1 - 2 \sin^2 A \cos^2 A}{1 - 2 \sin^2 A \cos^2 A} = 1
\]

Clearly, the left-hand side simplifies to 1:

\[
1 = 1
\]

Thus, the given equation is verified as true.

### Conclusion
The equation is indeed an identity.

---

Would you like further details or have any additional questions? Here are five related questions to consider:

1. How do we derive other trigonometric identities from basic ones like \(\sin^2 A + \cos^2 A = 1\)?
2. What happens if we encounter more complex trigonometric identities involving multiple angles?
3. How do we prove other similar identities involving powers of sine and cosine?
4. Can this identity be used in solving specific types of trigonometric equations?
5. How does the simplification of powers of trigonometric functions help in calculus?

**Tip:** Always look for standard trigonometric identities to simplify complex expressions—this approach can save time and reduce errors.

(sin^4 A + cos^4 A) / (1 - 2 sin^2 A cos^2 A) = 1

Learn how to prove the trigonometric identity (sin^4 A + cos^4 A) / (1 - 2 sin^2 A cos^2 A) = 1 by simplifying the equation using standard trigonometric identities. This detailed explanation is ideal for students in Grades 11-12.

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