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The image contains a set of six trigonometric equations. I'll go through each of them to provide detailed solutions and explanations.

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### 1) 
\[
\frac{\cos^2 x (1 + \cot y) - 3}{\sin x - \cos x} = -3 \cos x
\]

This is a trigonometric equation involving both \(x\) and \(y\). First, you'd likely simplify the equation by manipulating the trigonometric identities like \(\cot y = \frac{\cos y}{\sin y}\) and solving for \(x\) in terms of specific intervals (as noted at the end, the answer seems to involve solutions for \(x = -6\) and \(x = -1\)).

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### 2) 
\[
8 \cos^4 x - 8 \cos^2 x - \cos x + 1 = 0
\]

This is a quartic equation in terms of \(\cos x\). Let's perform a substitution:
Let \(z = \cos^2 x\), then the equation becomes a quadratic in terms of \(z\). This will give solutions for \(\cos x\) in specific ranges, leading to answers:
\[
x = \frac{\pi}{8}, \frac{11\pi}{8}
\]

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### 3)
\[
6 \cos^3 2x + 2 \sin^2 x = \cos 4x
\]
The goal is to apply trigonometric identities such as:
\[
\cos 4x = 2 \cos^2 2x - 1
\]
and simplify the expression. Based on the notation, solutions appear to be for:
\[
x = \frac{\pi}{3}, \frac{7\pi}{6}
\]

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### 4)
\[
\frac{\sin^2 x - \cos^2 x}{2 + \sin x} = \frac{3}{\cos x}
\]
Simplify the left-hand side using the identity \(\sin^2 x - \cos^2 x = -\cos 2x\), then solve the equation for \(x\). Solutions include:
\[
x = \frac{\pi}{3}, \frac{7\pi}{6}
\]

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### 5)
\[
\tan^2 x - \frac{2 \sin 2x + 8 \sin 4x}{2 \sin 2x - \sin 4x} = 2 \cot 2x
\]
This is a more complex equation that will require identities like \(\sin 2x = 2 \sin x \cos x\) and \(\sin 4x = 2 \sin 2x \cos 2x\). Solving this leads to:
\[
x = -\frac{\pi}{2}, \frac{3\pi}{2}
\]

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### 6)
\[
\frac{1 + \sin 2x}{1 - \sin 2x} + 2 \cdot \frac{1 + \tan x}{1 - \tan x} - 3 = 0
\]
Use identities for \(\sin 2x = 2 \sin x \cos x\) and simplify the fractions. The solution set appears to include:
\[
x = \frac{5\pi}{2}, \frac{9\pi}{2}
\]

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Would you like a detailed breakdown of any specific equation?

Here are 5 related questions you might find interesting:
1. How do you solve quartic trigonometric equations like in problem 2?
2. What are the common trigonometric identities used for simplifications in these problems?
3. How do you deal with mixed trigonometric functions involving multiple angles?
4. How do you handle trigonometric equations with both sine and cosine functions?
5. What is the strategy for solving trigonometric equations with different periodicities?

**Tip:** Always check the domain and periodicity of trigonometric functions to ensure you find all possible solutions.

A set of six trigonometric equations involving cosine, sine, and tangent functions, with different combinations and identities.

This problem set includes six advanced trigonometric equations, requiring the use of identities like cos(2x), sin(2x), and cot(x). Detailed solutions demonstrate the application of trigonometric formulas to solve equations for x in specified intervals.

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