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Let's work through the mathematical problems shown in the image. Here's a detailed explanation for each question.

### Problem 1:
You are given \(\omega = e^{\frac{2\pi i}{n}}\). The task is to show that:
\[
\prod_{k=0}^{n-1} \left( \alpha + \omega^k \right) = 2^n \prod_{k=0}^{n-1} \cos \left( \frac{\pi}{n} \right)
\]
This involves proving a product involving the powers of roots of unity \(\omega^k\), and linking this with a product of cosine terms.

**Solution Idea:**
- Recognize that \(\omega = e^{2\pi i / n}\) represents the roots of unity in the complex plane.
- Use trigonometric identities and properties of roots of unity to express the terms in the product, and simplify to a form involving cosine.

### Problem 2:
You are asked to calculate two sums:
\[
\sum_{k=0}^{n-1} \cos^2(kx + \varphi) \quad \text{and} \quad \sum_{k=0}^{n-1} \sin^2(kx + \varphi)
\]
These sums involve squares of cosine and sine functions for integer multiples of \(kx + \varphi\).

**Solution Idea:**
- Use the identity \(\cos^2 \theta + \sin^2 \theta = 1\) to simplify the sums.
- Recognize that both sums should have symmetry properties, potentially leading to simple expressions in terms of \(n\).

### Problem 3:
You are required to show that:
\[
\sum_{k=0}^{n} x^k \sin(k\theta) = \frac{x \sin \omega - x^{n+1} \sin((n+1)\theta) + x^{n+2} \sin(n\theta)}{1 - 2x \cos \theta + x^2}, \quad \text{where} \; x \in \mathbb{R}, \; \theta \neq k\pi
\]
This is a sum of terms involving powers of \(x\) multiplied by sine functions.

**Solution Idea:**
- Use geometric series summation techniques to handle the powers of \(x\).
- Apply known identities for sums involving sine functions to simplify the left-hand side and arrive at the given expression.

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These are the general solution approaches for each problem. Would you like a more detailed step-by-step breakdown of any specific problem? Feel free to ask for clarification or additional steps.

### Questions for further exploration:
1. How do roots of unity simplify complex products, such as in Problem 1?
2. Can we use Euler's formula \(e^{ix} = \cos x + i \sin x\) to solve these problems?
3. What are the properties of trigonometric sums when the arguments involve multiples of a common angle?
4. How do geometric series help in summing powers of a variable like in Problem 3?
5. What role do symmetry properties play in simplifying sums involving trigonometric functions?

### Tip:
For problems involving products or sums with roots of unity, it's useful to visualize the terms on the unit circle in the complex plane. This often leads to significant simplifications!

The query submitted by the user consists of three mathematical problems involving trigonometric sums, roots of unity, and series summation.

Explore detailed solutions to three problems involving trigonometric sums, roots of unity, and series expansion. This includes proving identities with cosine and sine functions, calculating complex products with roots of unity, and summing a geometric series involving trigonometric terms. Suitable for undergraduate math students.

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