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The image contains a series of mathematical statements related to trigonometric identities and calculus derivatives. Here's a detailed breakdown of the contents:

### Trigonometric Identity and Derivatives
1. **Given Trigonometric Identity:**
   \[
   \sin^2(x) + \cos^2(x) = 1
   \]

2. **Objective: Prove \(\sin(2x) = 2\sin(x)\cos(x)\)**

### Derivation Process:
- Start by using the double angle identity:
  \[
  \sin(2x) = 2\sin(x)\cos(x)
  \]

### Derivative Table:
The table appears to show standard derivatives for trigonometric functions.

| Function | Antiderivative |
|----------|----------------|
| \( \sin(x) \) | \( -\cos(x) \) |
| \( \cos(x) \) | \( \sin(x) \) |
| \( \tan(x) \) | \( -\ln|\cos(x)| \) |
| \( \sec^2(x) \) | \( \tan(x) \) |
| \( \csc^2(x) \) | \( -\cot(x) \) |
| \( \sec(x)\tan(x) \) | \( \sec(x) \) |
| \( \csc(x)\cot(x) \) | \( -\csc(x) \) |

### Exercises:
1. Differentiate \(\cos^2(x) + \sin^2(x)\):
   \[
   \frac{d}{dx}[\cos^2(x) + \sin^2(x)] = 0 \quad \text{(since it's a constant 1)}
   \]

2. Differentiate \(\sin(3x) = 3\sin(x)\cos(x)\):
   This seems to be an application of the chain rule. The correct derivative is:
   \[
   \frac{d}{dx}[\sin(3x)] = 3\cos(3x)
   \]
   
3. Differentiate \(\cos(2x) = \cos^2(x) - \sin^2(x)\):
   \[
   \frac{d}{dx}[\cos(2x)] = -2\sin(2x)
   \]

### Additional Notes:
- The questions and the table are designed to reinforce understanding of derivatives of basic trigonometric functions and identities.
- Ensure that when differentiating, the chain rule and trigonometric identities are used correctly.

Would you like more details on any specific part, or do you have any questions?

### Related Questions:
1. What is the derivative of \(\tan(x)\) with respect to \(x\)?
2. How can we derive the formula for the derivative of \(\sin^2(x)\)?
3. What are some applications of the double angle formula \(\sin(2x) = 2\sin(x)\cos(x)\) in calculus?
4. How does the chain rule apply when differentiating composite trigonometric functions?
5. What is the relationship between the derivatives of \(\sin(x)\) and \(\cos(x)\)?

### Tip:
Always remember that trigonometric identities can simplify differentiation and integration problems. Knowing these identities well can help you identify patterns and solve problems more efficiently.

Prove the trigonometric identity sin(2x) = 2sin(x)cos(x) and differentiate trigonometric functions.

Explore the proof of the trigonometric identity sin(2x) = 2sin(x)cos(x) and learn to differentiate various trigonometric functions. This problem covers key calculus concepts, including derivatives and the double angle identity. Suitable for students in Grades 11-12.

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