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The equation given is:

\[
\sin^2(3\alpha) - \cos^2(3\alpha) = -\cos(6\alpha)
\]

We can solve this step by step.

### Step 1: Use trigonometric identity
The expression on the left-hand side involves \(\sin^2(x) - \cos^2(x)\), which is a known identity:

\[
\sin^2(x) - \cos^2(x) = -\cos(2x)
\]

For \(x = 3\alpha\), this identity becomes:

\[
\sin^2(3\alpha) - \cos^2(3\alpha) = -\cos(6\alpha)
\]

### Step 2: Verify equality
Thus, the left-hand side is already equal to the right-hand side of the equation:

\[
-\cos(6\alpha) = -\cos(6\alpha)
\]

So the equation is **true** and holds for all \(\alpha\).

Would you like more details on any of the steps or additional clarifications?

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Here are 5 related questions to explore further:

1. How does the identity \(\sin^2(x) - \cos^2(x) = -\cos(2x)\) relate to other trigonometric identities?
2. Can you derive \(\sin^2(x) - \cos^2(x)\) from more fundamental trigonometric identities?
3. What are the applications of such trigonometric identities in real-world problems?
4. How does this identity compare to the more common \(\sin^2(x) + \cos^2(x) = 1\)?
5. Can you extend this identity to higher multiples of angles, like \(4x\) or \(5x\)?

**Tip**: Memorizing core trigonometric identities can significantly speed up solving complex trigonometric equations!

Solve the trigonometric equation sin^2(3α) - cos^2(3α) = -cos(6α) using angle identities. Learn step-by-step how to apply the double angle formula to simplify and verify the solution. Suitable for high school students in Grades 11-12.

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