https://file.aiquickdraw.com/mathbot/file/1726029733924chf2sjos.jpg

Let's go through the two questions presented in the image step by step.

### Question 12:
The expression given is:
\[
\left(1 - \sin^2(x)\right)^6 \cdot \left(\cos^2(x) - 1\right)^2
\]

We can use the Pythagorean identity to simplify this:

1. The identity \( 1 - \sin^2(x) = \cos^2(x) \), so the first part simplifies to:
   \[
   \left(\cos^2(x)\right)^6 = \cos^{12}(x)
   \]

2. Now simplify the second part \( \cos^2(x) - 1 \). This equals \( -\sin^2(x) \), so:
   \[
   \left(\cos^2(x) - 1\right)^2 = \left(-\sin^2(x)\right)^2 = \sin^4(x)
   \]

Thus, the whole expression simplifies to:
\[
\cos^{12}(x) \cdot \sin^4(x)
\]

So the correct answer is **B) \(\cos^{12}(x)\sin^4(x)\)**.

---

### Question 13:
The given information is \( \tan(x) = 16 \). We need to find the value of the expression:
\[
\frac{\sec(x)\sin(x)}{2\tan(x)\cot(x)} + \sqrt{\sec^2(x)-1}
\]

We proceed step by step:

1. Using the identity \( \sec^2(x) = 1 + \tan^2(x) \), we have:
   \[
   \sec^2(x) = 1 + 16^2 = 1 + 256 = 257
   \]

2. So, \( \sec(x) = \sqrt{257} \).

3. Next, simplify the term \( \sqrt{\sec^2(x) - 1} \):
   \[
   \sqrt{257 - 1} = \sqrt{256} = 16
   \]

4. Now simplify \( 2 \tan(x) \cot(x) \). Since \( \tan(x) \cdot \cot(x) = 1 \), we have:
   \[
   2 \tan(x) \cot(x) = 2
   \]

5. Now substitute into the original expression:
   \[
   \frac{\sec(x)\sin(x)}{2} + 16
   \]

6. Since \( \sin(x) = \frac{\tan(x)}{\sec(x)} \), we can find:
   \[
   \sin(x) = \frac{16}{\sqrt{257}}
   \]

7. Now substitute this into the expression:
   \[
   \frac{\sqrt{257} \cdot \frac{16}{\sqrt{257}}}{2} + 16 = \frac{16}{2} + 16 = 8 + 16 = 24
   \]

So, the correct answer is **A) 24**.

---

Would you like further clarification on any of these steps? Here are 5 related questions:

1. How do we derive trigonometric identities like \( \sin^2(x) + \cos^2(x) = 1 \)?
2. How does the Pythagorean identity help simplify trigonometric expressions?
3. What are common identities involving secant and tangent functions?
4. How can trigonometric values like \( \sin(x) \) be calculated given \( \tan(x) \)?
5. Why do \( \tan(x) \) and \( \cot(x) \) multiply to 1?

**Tip**: Understanding how to manipulate trigonometric identities is crucial for simplifying complex expressions efficiently.

Which of the following is equivalent to the expression given below? (1 - sin^2(x))^6 (cos^2(x) - 1)^2 

Given tan(x) = 16, find the value of the following expression: sec(x)sin(x) / [2tan(x)cot(x)] + sqrt(sec^2(x) - 1).

Learn how to simplify a complex trigonometric expression using Pythagorean identities and evaluate an expression with given tan(x) = 16. This problem covers advanced trigonometric formulas, including secant, tangent, sine, and cotangent functions, suitable for students in Grades 10-12.

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