prove the identity sin^2x+cos^2x=1 using pythagoras and the unit cirlce diagram

pls explain this question and solve it simply

The identity \( \sin^2x + \cos^2x = 1 \) is a fundamental trigonometric identity. To prove this identity, we can use the concepts of the unit circle and the Pythagorean theorem. Let's break it down step by step.

### Step 1: Understanding the Unit Circle
The unit circle is a circle with a radius of 1, centered at the origin (0, 0) on the coordinate plane. Any point on the unit circle has coordinates \((\cos x, \sin x)\), where \(x\) is the angle formed with the positive x-axis.

### Step 2: Coordinates on the Unit Circle
If you pick any angle \(x\) on the unit circle, the coordinates of the corresponding point on the circle are given by:
\[
(\cos x, \sin x)
\]
This means the horizontal distance from the origin to the point is \(\cos x\) (adjacent side), and the vertical distance is \(\sin x\) (opposite side).

### Step 3: Applying the Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs (adjacent and opposite sides) is equal to the square of the length of the hypotenuse. For our case:
\[
(\text{Adjacent side})^2 + (\text{Opposite side})^2 = (\text{Hypotenuse})^2
\]
Here, the hypotenuse is the radius of the unit circle, which is 1. So we have:
\[
\cos^2 x + \sin^2 x = 1^2
\]

### Step 4: Conclusion
This equation simplifies to:
\[
\cos^2 x + \sin^2 x = 1
\]
This is exactly the identity we wanted to prove!

### Visual Understanding
- Imagine a right triangle inscribed within the unit circle.
- The horizontal leg is \(\cos x\), the vertical leg is \(\sin x\), and the hypotenuse is the radius of the unit circle (which is 1).
- By the Pythagorean theorem, \(\cos^2 x + \sin^2 x\) must equal 1.

### Summary
Using the unit circle and the Pythagorean theorem, we've demonstrated that the identity \( \sin^2x + \cos^2x = 1 \) is true for any angle \(x\).

Would you like more details or have any questions?

Here are five related questions to further your understanding:

1. How does the identity \( \sin^2x + \cos^2x = 1 \) apply to different angles, like \(x = 0^\circ\), \(90^\circ\), or \(45^\circ\)?
2. How would the identity change if the circle were not a unit circle but had a different radius?
3. Can you explain how the sine and cosine functions relate to each other using the unit circle?
4. How does the identity \( \sin^2x + \cos^2x = 1 \) help in solving trigonometric equations?
5. What are other fundamental trigonometric identities derived from this one?

**Tip:** The unit circle is a powerful tool for understanding trigonometric identities, and practicing with it can make solving trigonometric problems much easier.

Prove the identity sin^2x + cos^2x = 1 using Pythagoras and the unit circle diagram. Please explain this question and solve it simply.

Learn how to prove the fundamental trigonometric identity sin^2x + cos^2x = 1 using the Pythagorean Theorem and the unit circle. This explanation is designed for students in Grades 9-11, covering key trigonometric concepts and providing a step-by-step solution.

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