The trigonometric identity involving the sum of the squares of the sine and cosine functions is:

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

This is known as the Pythagorean identity, which is fundamental in trigonometry.

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Here are some related questions you might be interested in:

1. How is the Pythagorean identity derived?
2. Can you prove the Pythagorean identity using the unit circle?
3. What are other trigonometric identities involving \(\sin(x)\) and \(\cos(x)\)?
4. How can the Pythagorean identity be used to simplify trigonometric expressions?
5. What is the relationship between the Pythagorean identity and the Pythagorean theorem?
6. How can the Pythagorean identity be applied in solving trigonometric equations?
7. What are the co-function identities in trigonometry?
8. How does the Pythagorean identity change for different angles, such as \(x = 90^\circ\) or \(x = 45^\circ\)?

**Tip:** Trigonometric identities, like \(\cos^2(x) + \sin^2(x) = 1\), are very useful in simplifying complex trigonometric expressions and solving trigonometric equations. Memorizing these identities can greatly aid in your studies of trigonometry.

Explore the fundamental trigonometric identity \( \cos^2(x) + \sin^2(x) = 1 \) and its implications in trigonometry. This identity simplifies trigonometric expressions and is essential for solving trigonometric equations. Suitable for high school students and anyone studying trigonometry.

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