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Let's tackle the problems one by one:

### Problem 10: Simplify \(\frac{\sin(x) + \cos(x)}{\tan(x) \cdot \cot(x)}\)

First, let's simplify the denominator. We know that:
\[
\tan(x) = \frac{\sin(x)}{\cos(x)} \quad \text{and} \quad \cot(x) = \frac{\cos(x)}{\sin(x)}
\]

Thus, the product of \(\tan(x)\) and \(\cot(x)\) is:
\[
\tan(x) \cdot \cot(x) = \left(\frac{\sin(x)}{\cos(x)}\right) \cdot \left(\frac{\cos(x)}{\sin(x)}\right) = 1
\]

This simplifies the expression to:
\[
\frac{\sin(x) + \cos(x)}{1} = \sin(x) + \cos(x)
\]

Thus, the correct answer is **B) \(\cos(x) + \sin(x)\)**.

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### Problem 11: Which of the following is equivalent to the expression \(\tan^4(\theta) + 2\tan^2(\theta) + 1\)?

Let’s analyze the given expression:

\[
\tan^4(\theta) + 2\tan^2(\theta) + 1
\]

Notice that this can be factored as:
\[
(\tan^2(\theta) + 1)^2
\]

Now, using the identity:
\[
\tan^2(\theta) + 1 = \sec^2(\theta)
\]

We substitute this into the expression:
\[
(\sec^2(\theta))^2 = \sec^4(\theta)
\]

Thus, the correct answer is **C) \(\sec^4(\theta)\)**.

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Do you need more details or have any questions about the steps?

Here are five related questions for further practice:

1. Simplify \(\frac{\sin(x) \cdot \cos(x)}{\tan(x)}\).
2. If \(\tan^2(x) + 1 = \sec^2(x)\), what is \(\cot^2(x) + 1\) in terms of \(\csc(x)\)?
3. Simplify \(\frac{\sin(x)}{\cos(x)} + \frac{\cos(x)}{\sin(x)}\).
4. How can the expression \(\sec^2(x) - \tan^2(x)\) be simplified?
5. Factor the expression \(\sin^4(x) + 2\sin^2(x) + 1\).

### Tip:
Always try to recognize common trigonometric identities during simplifications, as they often lead to faster solutions!

Problem 10: Simplify (sin(x) + cos(x)) / (tan(x) * cot(x)) and Problem 11: Which of the following is equivalent to tan^4(θ) + 2tan^2(θ) + 1?

Learn how to simplify trigonometric expressions like (sin(x) + cos(x)) / (tan(x) * cot(x)) and find the equivalent form of tan^4(θ) + 2tan^2(θ) + 1. This guide covers essential trigonometric identities and their applications for high school students.

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