The expression \( \sin^2(\theta) + \cos^2(\theta) \) simplifies to 1, based on the **Pythagorean identity** in trigonometry:

\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]

This identity holds for all values of \( \theta \).

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:

1. How is the Pythagorean identity derived from the unit circle?
2. What is the relationship between \( \sin(\theta) \) and \( \cos(\theta) \) for complementary angles?
3. How does \( \sin^2(\theta) + \cos^2(\theta) = 1 \) help in solving trigonometric equations?
4. What are the other Pythagorean identities involving \( \tan(\theta) \) and \( \sec(\theta) \)?
5. How does this identity apply in solving right triangles?

**Tip:** Remember that this identity is foundational in trigonometry and can help simplify complex expressions in calculus and physics.

Understand how the trigonometric identity sin^2(θ) + cos^2(θ) simplifies to 1. This foundational concept is essential in trigonometry and is derived from the Pythagorean theorem, suitable for students in Grades 9-12.

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