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I can help you with the math problems in the image. Here’s a breakdown of each equation:

1. **\( \frac{\cos^2 x (1 + \cot y) - 3}{\sin x - \cos x} = -3 \cos x \)**
   - The solution provided appears to be: \( x = 6 \) or \( x = -1 \).

2. **\( 8 \cos^4 x - 8 \cos^2 x - \cos x + 1 = 0 \)**
   - The solutions seem to be: \( x = \frac{\pi}{3}, \frac{11\pi}{6} \).

3. **\( 6 \cos^3 2x + 2 \sin^3 2x = \cos 4x \)**
   - The solution appears to be \( x = \frac{2\pi}{3}, \frac{\pi}{6} \).

4. **\( \frac{\sin^2 x - \cos^2 x}{2 + \sin x} = \frac{3 \cos x}{2} \)**
   - The solution is indicated as \( x = -\frac{\pi}{3}, \frac{7\pi}{6} \).

5. **\( \frac{\tan^2 x - 2 \sin 2x + 8 \sin 4x}{2 \sin 2x - \sin 4x} = 2 \cot^2 x \)**
   - The solution shown is \( x = \frac{3}{2} \pi, \frac{\pi}{4} \).

6. **\( \frac{1 + \sin 2x}{1 - \sin 2x} + 2 \frac{1 + \tan x}{1 - \tan x} - 3 = 0 \)**
   - The solutions given are \( x = \frac{5\pi}{2}, \frac{9\pi}{4} \).

If you would like a detailed explanation of any specific equation or steps involved in solving them, feel free to ask!

Do you have any questions about these equations? Also, here are five related questions to help deepen your understanding:

1. How do you solve trigonometric equations involving multiple angles?
2. What are the common trigonometric identities used in simplifying these types of equations?
3. Can you explain how to convert between trigonometric forms (e.g., using identities)?
4. How would you handle equations with both sine and cosine terms?
5. Why is it important to check for extraneous solutions in trigonometric equations?

**Tip:** Always verify the domain of the solution when working with trigonometric equations, as some solutions might fall outside the allowed interval.

Solve the set of trigonometric equations displayed in the image, which involve various trigonometric identities and algebraic manipulations.

Explore the solution to a set of trigonometric equations using key trigonometric identities, including the Pythagorean, tangent, and double-angle formulas. This problem is designed for high school and college-level students studying advanced trigonometry.

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