
Expand
1. sin(2a) 
2. cos(2a) 
3.  tan(a+b) 
4. tan(a-b) 

Simplify
5. [sin²(a) ]/[1-cos²(a)]
6. sec²(a) - tan²(a)

### Expansion

1. **\(\sin(2a)\):**
   \[
   \sin(2a) = 2\sin(a)\cos(a)
   \]

2. **\(\cos(2a)\):**  
   There are multiple equivalent forms, depending on what you're working with:
   \[
   \cos(2a) = \cos^2(a) - \sin^2(a)
   \]
   Alternatively, using trigonometric identities:
   \[
   \cos(2a) = 2\cos^2(a) - 1
   \]
   or
   \[
   \cos(2a) = 1 - 2\sin^2(a)
   \]

3. **\(\tan(a + b)\):**
   \[
   \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}
   \]

4. **\(\tan(a - b)\):**
   \[
   \tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}
   \]

### Simplifications

5. **\(\frac{\sin^2(a)}{1 - \cos^2(a)}\):**  
   Using the identity \(1 - \cos^2(a) = \sin^2(a)\), we get:
   \[
   \frac{\sin^2(a)}{1 - \cos^2(a)} = \frac{\sin^2(a)}{\sin^2(a)} = 1
   \]

6. **\(\sec^2(a) - \tan^2(a)\):**  
   Using the Pythagorean identity \(\sec^2(a) = 1 + \tan^2(a)\), we have:
   \[
   \sec^2(a) - \tan^2(a) = (1 + \tan^2(a)) - \tan^2(a) = 1
   \]

Would you like more details or have any further questions?

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### Related Questions:

1. What are the other forms of \(\sin(2a)\) using different identities?
2. How does the expansion of \(\cos(2a)\) relate to double-angle formulas?
3. Can you derive the formula for \(\tan(a + b)\) from basic trigonometric principles?
4. How do Pythagorean identities help in simplifying trigonometric expressions?
5. What is the geometric interpretation of \(\tan(a - b)\)?

### Tip:
Always remember that trigonometric identities are powerful tools for simplifications. Keep the core ones like Pythagorean, double-angle, and sum/difference formulas handy!

Expand: 1. sin(2a) 2. cos(2a) 3. tan(a+b) 4. tan(a-b) Simplify: 5. [sin²(a)]/[1-cos²(a)] 6. sec²(a) - tan²(a)

Explore detailed solutions for expanding and simplifying trigonometric identities such as sin(2a), cos(2a), tan(a+b), and tan(a-b). This guide covers key formulas, including double-angle and sum/difference identities, making it perfect for high school students in Grades 10-12.

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