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### Formula:
\[
\sin^2 x + \cos^2 x = 1
\]

A table is provided for different values of \( x \), \( \sin x \), \( \cos x \), and \( \sin^2 x + \cos^2 x \), which confirms the identity by computing the squares of sine and cosine for specific angles.

- **x = 0°**
  - \( \sin(0°) = 0 \)
  - \( \cos(0°) = 1 \)
  - \( \sin^2(0°) + \cos^2(0°) = 0^2 + 1^2 = 1 \)

- **x = 30°**
  - \( \sin(30°) = \frac{1}{2} \)
  - \( \cos(30°) = \frac{\sqrt{3}}{2} \)
  - \( \sin^2(30°) + \cos^2(30°) = \left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = 1 \)

- **x = 45°**
  - \( \sin(45°) = \frac{\sqrt{2}}{2} \)
  - \( \cos(45°) = \frac{\sqrt{2}}{2} \)
  - \( \sin^2(45°) + \cos^2(45°) = \left(\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2 = 1 \)

- **x = 60°**
  - \( \sin(60°) = \frac{\sqrt{3}}{2} \)
  - \( \cos(60°) = \frac{1}{2} \)
  - \( \sin^2(60°) + \cos^2(60°) = \left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2 = 1 \)

The results confirm the identity \( \sin^2 x + \cos^2 x = 1 \) for the angles tested.

Would you like to dive deeper into any particular trigonometric identities or solve similar problems?

### Questions:
1. How do we derive the identity \( \sin^2 x + \cos^2 x = 1 \) from the Pythagorean theorem?
2. Can the same identity be extended to complex numbers?
3. What happens if we substitute other trigonometric values for sine and cosine, like tangent or secant?
4. How would this identity behave for negative angles or angles beyond 360°?
5. What other fundamental trigonometric identities arise from \( \sin^2 x + \cos^2 x = 1 \)?

### Tip:
Remember, verifying identities using specific angles can help solidify understanding, but proving them algebraically ensures they hold universally.

Prove the trigonometric identity \( \sin^2 x + \cos^2 x = 1 \) for different angles and values of sine and cosine.

This guide walks through proving the trigonometric identity \( \sin^2 x + \cos^2 x = 1 \) using sine and cosine values at specific angles such as 0°, 30°, 45°, and 60°. Step-by-step validation confirms the identity for these common angles, which is essential for high school trigonometry students.

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