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The image contains a math worksheet with trigonometric identities and equations to prove or solve. Let's walk through the key exercises:

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### 1. **Exercise 1**
   **(a)**: Prove that the equation \( \cos 2x = \sin x \) can be rewritten as \( \sin^2 x + \sin x - 1 = 0 \).  
   **(b)**: Solve \( 2x = \sin x \) in the interval \( -\pi \leq x \leq \pi \).

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### 2. **Solve** \( 4 \sin 3x = \frac{1}{\cos 3x} \) for \( 0 \leq x \leq \pi \).

This can be simplified by rewriting in terms of trigonometric identities.

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### 3. **Solve** \( \cos^2 x - 3 \sin^2 x = 0 \) for \( 0 \leq x \leq \pi \).  
Use the identity \( \cos^2 x + \sin^2 x = 1 \).

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### 4. **Solve** \( \cos 2x - \cos^2 x = 0 \) for \( 0 \leq x \leq 3\pi \).

Expand \( \cos 2x \) using the double-angle formula:  
\( \cos 2x = 2 \cos^2 x - 1 \).

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### 5. **Exercise 5**
   **(a)**: Prove that \( \sin 2x + \cos 2x - 1 = 2 \sin x (\cos x - \sin x) \).  
   **(b)**: Solve \( \sin 2x + \cos 2x - 1 + \cos x - \sin x = 0 \) for \( 0 < x < 2\pi \).

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### 6. **Exercise 6**  
   **(a)**: Show that \( 2 \cos^2 x + 5 \sin x = 4 \) can be rewritten as \( 2 \sin^2 x - 5 \sin x + 2 = 0 \).  
   **(b)**: Solve the equation \( 2 \cos^2 x + 5 \sin x = 4 \) for \( 0 \leq x \leq 2\pi \).

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### 7. **Solve** \( 2 \cos^2 x + 5 \sin x = 4 \) for \( 0 \leq x \leq 2\pi \).  

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Would you like me to solve a specific part or provide step-by-step explanations for one or more of these exercises? Let me know!

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Here are **5 related questions** you might explore:
1. How can trigonometric identities simplify complex trigonometric equations?
2. What are the key properties of the double-angle identities?
3. How does squaring both sides of trigonometric equations affect the solution?
4. What strategies can help avoid extraneous solutions in trigonometric equations?
5. What is the geometric meaning of solutions to \( \cos 2x = \sin x \)?

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**Tip:** Always check the domain and possible extraneous solutions when dealing with trigonometric identities and equations.

Solve various trigonometric equations and prove trigonometric identities.

Learn how to solve various trigonometric equations and prove trigonometric identities with step-by-step solutions. This worksheet covers key topics such as double-angle formulas, trigonometric simplification, and solving equations over specified intervals. Suitable for high school students in Grades 10-12.

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