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I see you've uploaded a handwritten document containing trigonometric identities and problems. Let me extract the content and provide solutions step by step.

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### Section I: Expand the expressions
1. **Expand \( \cos^2 a - 2 \cos a \cos m \)**
   Using trigonometric identities:
   - \( \cos^2 a = \frac{1 + \cos(2a)}{2} \)
   - \( 2 \cos a \cos m = \cos(a + m) + \cos(a - m) \)

   Expanded form:
   \[
   \cos^2 a - 2 \cos a \cos m = \frac{1 + \cos(2a)}{2} - \cos(a + m) - \cos(a - m)
   \]

2. **Expand \( \cos^2 n - 2 \sin m \cos m \)**
   - \( \cos^2 n = \frac{1 + \cos(2n)}{2} \)
   - \( 2 \sin m \cos m = \sin(2m) \)

   Expanded form:
   \[
   \cos^2 n - 2 \sin m \cos m = \frac{1 + \cos(2n)}{2} - \sin(2m)
   \]

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### Section II: Verify the identity
You are tasked with verifying trigonometric equations, including:
\[
\sin(A + B) + \sin(A - B) = 2 \sin A \cos B
\]
This is a standard identity:
- Left-hand side (LHS):
  \[
  \sin(A + B) + \sin(A - B) = 2 \sin A \cos B
  \]
- Verified: LHS = RHS.

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### Section III: Verify identities
1. **Verify \( \sin(3a) = 3 \sin a - 4 \sin^3 a \):**
   Using the triple angle identity:
   - \( \sin(3a) = 3 \sin a - 4 \sin^3 a \)
   - LHS = RHS (standard formula for \( \sin(3a) \)).

2. **Verify \( \tan\left(\frac{A}{2}\right) = \frac{\sin A}{1 + \cos A} \):**
   Using half-angle identities:
   - \( \tan\left(\frac{A}{2}\right) = \frac{1 - \cos A}{\sin A} \) (reciprocal).

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### Section IV: Solve equations
1. \( \sin a \cos a = \frac{\sqrt{2}}{8} \):
   - Use the identity \( \sin a \cos a = \frac{1}{2} \sin(2a) \):
     \[
     \frac{1}{2} \sin(2a) = \frac{\sqrt{2}}{8} \implies \sin(2a) = \frac{\sqrt{2}}{4}
     \]
     Solve \( 2a \) using arcsin.

2. \( \tan^2(3a) + \sin^2(6a) = 2 \):
   Rearrange and solve using trigonometric squares.

3. \( \tan(2a) + \tan(3a) + \tan(3a) = 0 \):
   Use the tangent addition formula.

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### Section V: Simplify the expression
For \( P = \cos^2 \alpha + 2 \cos \beta \cos \theta - 1 \):
- Simplify each term using trigonometric squares and identities.

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Would you like detailed solutions for each problem? Let me know!  

Here are five questions to deepen your understanding:
1. How can you derive the double-angle identity for sine and cosine?
2. What are the steps to simplify trigonometric equations involving products?
3. Why is it helpful to express \( \cos^2 a \) in terms of \( \cos(2a) \)?
4. How does verifying identities help in solving complex trigonometric equations?
5. What is the significance of using half-angle formulas?

**Tip:** Always simplify expressions by substituting known identities before expanding or solving equations!

The document contains a set of trigonometric problems and identities for expansion, verification, and simplification.

Explore step-by-step solutions to trigonometric problems involving expansions, identity verification, and simplifications. Learn formulas such as double-angle and sum-difference identities, suitable for high school math students in Grades 10-12.

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