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To solve the given problems, we need to analyze the given information and apply appropriate trigonometric identities and properties of triangles.

### Given:
\[
\text{In } \triangle ABC, \text{ with angles } A, B, \text{ and } C \text{, and opposite sides } a, b, \text{ and } c \text{ respectively, we have:}
\]
\[
\frac{\cos A}{1 + \sin A} = \frac{\sin 2B}{1 + \cos 2B}
\]

### Part 1: Find angle \( B \) given \( C = \frac{2\pi}{3} \)

Using the identity for \( \sin 2B \) and \( \cos 2B \):
\[
\sin 2B = 2 \sin B \cos B
\]
\[
\cos 2B = 2 \cos^2 B - 1
\]

Substituting these into the given equation:
\[
\frac{\cos A}{1 + \sin A} = \frac{2 \sin B \cos B}{1 + (2 \cos^2 B - 1)}
\]
\[
\frac{\cos A}{1 + \sin A} = \frac{2 \sin B \cos B}{2 \cos^2 B}
\]
\[
\frac{\cos A}{1 + \sin A} = \frac{\sin B}{\cos B}
\]
\[
\frac{\cos A}{1 + \sin A} = \tan B
\]

Since \( C = \frac{2\pi}{3} \), the sum of the angles in \( \triangle ABC \) is \( \pi \):
\[
A + B + \frac{2\pi}{3} = \pi
\]
\[
A + B = \frac{\pi}{3}
\]

### Part 2: Find the minimum value of \( \frac{a^2 + b^2}{c^2} \)

Using the Law of Cosines in \( \triangle ABC \):
\[
c^2 = a^2 + b^2 - 2ab \cos C
\]

Given \( C = \frac{2\pi}{3} \):
\[
\cos C = \cos \left(\frac{2\pi}{3}\right) = -\frac{1}{2}
\]
\[
c^2 = a^2 + b^2 - 2ab \left(-\frac{1}{2}\right)
\]
\[
c^2 = a^2 + b^2 + ab
\]

To find the minimum value of \( \frac{a^2 + b^2}{c^2} \):
\[
\frac{a^2 + b^2}{c^2} = \frac{a^2 + b^2}{a^2 + b^2 + ab}
\]

This ratio achieves its minimum value when \( a \) and \( b \) are equal, i.e., \( a = b \):
\[
\frac{a^2 + b^2}{a^2 + b^2 + ab} = \frac{2a^2}{2a^2 + a^2} = \frac{2}{3}
\]

Thus, the minimum value of \( \frac{a^2 + b^2}{c^2} \) is:
\[
\boxed{\frac{2}{3}}
\]

Would you like further details on this solution or have any other questions?

Here are 5 questions you might want to ask:
1. How do you apply the Law of Cosines in different triangles?
2. Can you explain the derivation of trigonometric identities?
3. What are the properties of angles in a triangle?
4. How can you verify the minimum value of a trigonometric expression?
5. What are some real-life applications of triangle properties?

**Tip:** When solving problems involving trigonometric identities, always simplify the equation step-by-step and substitute known values to reduce complexity.

Explore the solution involving triangle properties and trigonometric identities. Understand how to apply the Law of Cosines and analyze the sum of angles in a triangle. Suitable for advanced high school students (Grades 10-12).

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