To solve the equation \( \sin^4 A - \cos^4 A = 1 - 2\cos^2 A \), let’s proceed with simplification and verification of each side.

### Step 1: Expand the Left Side

The left side, \( \sin^4 A - \cos^4 A \), is a difference of squares:
\[
\sin^4 A - \cos^4 A = (\sin^2 A + \cos^2 A)(\sin^2 A - \cos^2 A)
\]
Since \( \sin^2 A + \cos^2 A = 1 \) (by the Pythagorean identity), this becomes:
\[
\sin^4 A - \cos^4 A = (1)(\sin^2 A - \cos^2 A) = \sin^2 A - \cos^2 A
\]

### Step 2: Express \( \sin^2 A \) in Terms of \( \cos^2 A \)

Using the identity \( \sin^2 A = 1 - \cos^2 A \), we can rewrite \( \sin^2 A - \cos^2 A \) as:
\[
\sin^2 A - \cos^2 A = (1 - \cos^2 A) - \cos^2 A = 1 - 2\cos^2 A
\]

### Step 3: Substitute and Verify

Now, we see that:
\[
\sin^4 A - \cos^4 A = 1 - 2\cos^2 A
\]

This matches the right side, confirming that the equation is an identity:
\[
\sin^4 A - \cos^4 A = 1 - 2\cos^2 A
\]

This equation is indeed true.

Would you like more details, or do you have any questions?

### Follow-Up Questions

1. Can you apply this approach to verify other trigonometric identities?
2. How would you expand and simplify if \( \sin^4 A + \cos^4 A \) were given?
3. Can you prove similar identities involving higher powers of sine and cosine?
4. What is the importance of the Pythagorean identity in trigonometric simplifications?
5. Can you use double-angle identities to further explore similar identities?

### Tip

When verifying trigonometric identities, try breaking down complex expressions into known identities (like the Pythagorean identity) to simplify each side step-by-step.

Learn how to simplify the trigonometric identity sin^4 A - cos^4 A to 1 - 2cos^2 A, using key concepts like the Pythagorean identity and the difference of squares. This step-by-step solution is suitable for students in Grades 9-11.

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