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The given equation is:

\[
\sin^6 \theta + \cos^6 \theta + 3 \sin^2 \theta \cos^2 \theta = 1
\]

Let's simplify and verify the equation:

### Step 1: Use identity for \(\sin^2 \theta + \cos^2 \theta = 1\)
We can attempt to transform the terms by recognizing standard trigonometric identities. One useful identity here is the following:

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

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

\[
\sin^6 \theta + \cos^6 \theta = 1 - 3 \sin^2 \theta \cos^2 \theta
\]

### Step 2: Substitute into the original equation
Now substitute this into the original equation:

\[
1 - 3 \sin^2 \theta \cos^2 \theta + 3 \sin^2 \theta \cos^2 \theta = 1
\]

### Step 3: Simplification
Notice that the terms \( - 3 \sin^2 \theta \cos^2 \theta \) and \( + 3 \sin^2 \theta \cos^2 \theta \) cancel out:

\[
1 = 1
\]

This is a true statement, meaning the equation is **identically true** for all values of \(\theta\).

### Conclusion
The given equation holds true for all values of \(\theta\).

Would you like further explanation or details?

Here are 5 related questions for deeper understanding:
1. What are the key trigonometric identities used in simplifying expressions?
2. How do we derive the identity \(\sin^6 \theta + \cos^6 \theta\)?
3. Can you explain the significance of the identity \(\sin^2 \theta + \cos^2 \theta = 1\)?
4. What happens when you substitute specific angles (like \(\theta = 0\) or \(\theta = \frac{\pi}{4}\)) into the equation?
5. How would this equation change if there were additional powers or terms?

**Tip:** Trigonometric identities are often the key to simplifying complex trigonometric equations, so practice deriving and applying them frequently.

sin^6 θ + cos^6 θ + 3 sin^2 θ cos^2 θ = 1

Learn how to simplify and verify the trigonometric equation sin^6 θ + cos^6 θ + 3 sin^2 θ cos^2 θ = 1 using key identities such as sin^2 θ + cos^2 θ = 1. This solution involves simplification and substitution for deeper understanding. Suitable for students in Grades 10-12.

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