Given sin(𝜃𝜃) = 15 and cos(𝛽𝛽) = 1, where 𝜋𝜋 < 𝜃𝜃 < π and 3𝜋𝜋 < 𝛽𝛽 < 2π, determine the exact value of cos(𝜃𝜃 + 𝛽𝛽). 17 3 2 2
6. Simplify each expression to a single trigonometric ratio.
𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝜋 b. sin(5)sin(3) − cos(5)cos(3)
a. sin(3x)cos(x) − cos(3x)sin(x)
c.
1−tan(800) tan(200) tan(800)+tan(200)

Let's break down and solve each part of your problem step by step.

### Part 1: Determine the exact value of \( \cos(\theta + \beta) \)

#### Given:
- \( \sin(\theta) = 15 \) with \( \pi < \theta < \pi \)
- \( \cos(\beta) = 1 \) with \( 3\pi < \beta < 2\pi \)

### Step 1: Analyze the given conditions
1. **For \( \sin(\theta) = 15 \):** The sine function ranges between \(-1\) and \(1\), so \( \sin(\theta) = 15 \) is not possible, indicating that there might be a typo. Please verify the value.

2. **For \( \cos(\beta) = 1 \):**
   - \( \cos(\beta) = 1 \) occurs when \( \beta = 2n\pi \) for any integer \( n \).
   - However, since \( 3\pi < \beta < 2\pi \) is given, \( \cos(\beta) = 1 \) at \( \beta = 2\pi \).

Given that \( \sin(\theta) = 15 \) is likely a mistake, please confirm or correct this value so I can accurately calculate \( \cos(\theta + \beta) \).

### Part 2: Simplify each expression to a single trigonometric ratio.

#### (a) Simplify \( \sin(3x)\cos(x) - \cos(3x)\sin(x) \)

This expression resembles the sine angle subtraction identity:

\[
\sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B)
\]

Let \( A = 3x \) and \( B = x \). Thus:

\[
\sin(3x)\cos(x) - \cos(3x)\sin(x) = \sin(3x - x) = \sin(2x)
\]

**Answer:** \( \sin(2x) \)

#### (b) Simplify \( \sin(5)\sin(3) - \cos(5)\cos(3) \)

This expression resembles the cosine angle addition identity:

\[
\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B)
\]

Here, let \( A = 5 \) and \( B = 3 \):

\[
\sin(5)\sin(3) - \cos(5)\cos(3) = -\cos(5 + 3) = -\cos(8)
\]

**Answer:** \( -\cos(8) \)

#### (c) Simplify \( \frac{1 - \tan(800) \tan(200)}{\tan(800) + \tan(200)} \)

This expression resembles the tangent angle subtraction identity:

\[
\tan(A - B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)}
\]

Rewriting the given expression:

\[
\frac{1 - \tan(800^\circ)\tan(200^\circ)}{\tan(800^\circ) + \tan(200^\circ)} = \tan(1000^\circ - 600^\circ)
\]

Simplify:

\[
= \tan(400^\circ)
\]

Since \( \tan(400^\circ) = \tan(400^\circ - 360^\circ) = \tan(40^\circ) \):

**Answer:** \( \tan(40^\circ) \)

### Summary:
1. For \( \cos(\theta + \beta) \), we need the correct value for \( \sin(\theta) \) to proceed.
2. \( \sin(3x)\cos(x) - \cos(3x)\sin(x) = \sin(2x) \)
3. \( \sin(5)\sin(3) - \cos(5)\cos(3) = -\cos(8) \)
4. \( \frac{1 - \tan(800^\circ) \tan(200^\circ)}{\tan(800^\circ) + \tan(200^\circ)} = \tan(40^\circ) \)

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Would you like more details or clarifications? If so, please specify which part you'd like to discuss further.

Here are some related questions:

1. What is the sum-to-product identity for sine and cosine?
2. How do we use angle identities to simplify expressions?
3. Can you calculate exact values for trigonometric functions without a calculator?
4. What happens if the argument of a tangent function exceeds 360 degrees?
5. What is the significance of the range of trigonometric functions?
6. How can errors in trigonometric problem setups be identified and corrected?
7. What is the geometric interpretation of sine and cosine on the unit circle?
8. How do the angle addition and subtraction formulas for sine and cosine work?

**Tip:** Always check the domain restrictions and ranges of trigonometric functions when dealing with angle problems.

Learn how to solve advanced trigonometric problems involving sine, cosine, and tangent functions. This detailed solution covers angle addition and subtraction identities, range and domain considerations, and specific examples like calculating cosine of a sum of angles. Suitable for advanced high school students and beyond.

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