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) $\cos(x) + \sin(x)$.

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) $\sec^4(\theta)$.

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!