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Let's analyze and simplify the given expression:

\[
\sin^6 \theta + \cos^6 \theta
\]

### Step 1: Use the identity for \(a^3 + b^3\)
The expression can be rewritten as:
\[
\sin^6 \theta + \cos^6 \theta = (\sin^2 \theta)^3 + (\cos^2 \theta)^3
\]

We know that:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
Let \(a = \sin^2 \theta\) and \(b = \cos^2 \theta\). Applying this identity gives:

\[
\sin^6 \theta + \cos^6 \theta = (\sin^2 \theta + \cos^2 \theta)\left[(\sin^2 \theta)^2 - \sin^2 \theta \cdot \cos^2 \theta + (\cos^2 \theta)^2\right]
\]

### Step 2: Simplify using the Pythagorean identity
Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\), we have:

\[
\sin^6 \theta + \cos^6 \theta = 1 \cdot \left[(\sin^4 \theta - \sin^2 \theta \cos^2 \theta + \cos^4 \theta)\right]
\]

This simplifies to:

\[
\sin^6 \theta + \cos^6 \theta = \sin^4 \theta - \sin^2 \theta \cos^2 \theta + \cos^4 \theta
\]

### Step 3: Further factorization
The expression can be further simplified using the identity:
\[
\sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2\sin^2 \theta \cos^2 \theta
\]
Substituting this, we get:
\[
\sin^6 \theta + \cos^6 \theta = [1 - 2\sin^2 \theta \cos^2 \theta] - \sin^2 \theta \cos^2 \theta = 1 - 3\sin^2 \theta \cos^2 \theta
\]

### Final Answer:
\[
\sin^6 \theta + \cos^6 \theta = 1 - 3\sin^2 \theta \cos^2 \theta
\]

This is the simplified form of the expression.

Would you like to ask anything else, or see any related examples?

Here are some related questions:

1. How do you expand and simplify \((\sin^2 \theta + \cos^2 \theta)^3\)?
2. What is the significance of the identity \(\sin^2 \theta + \cos^2 \theta = 1\)?
3. Can you factor the expression \(\sin^4 \theta - \cos^4 \theta\)?
4. What is the geometric interpretation of \(\sin^2 \theta\) and \(\cos^2 \theta\)?
5. How would the expression change if you used \(\tan \theta\) instead?

**Tip:** When dealing with expressions that involve powers of sine and cosine, try to use known identities and factorization techniques to simplify them efficiently.

Learn how to simplify the expression sin^6 theta + cos^6 theta step-by-step using trigonometric identities and factorization techniques. This problem covers key trigonometric concepts, including the sum of cubes identity and the Pythagorean identity. Suitable for advanced high school students and enthusiasts of trigonometry.

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