I need help proving the identity of the statement and the rule ### Proving the Trigonometric Identity#### Objective:Prove the following trigonometric identity:\[ \tan(2\pi - x) = -\tan(x) \]#### Instructions:Each statement in your proof must be based on a Rule selected from the Rule menu. For additional details about a Rule, select the "More Information" button on the right of the Rule.---#### Steps:1. **Statement and Rule Interface:** - **Statement:** \[ \tan(2\pi - x) \] - **Rule:** Select an appropriate trigonometric rule from the drop-down menu.2. **Interface Options:** - The interface provides a list of trigonometric functions and constants for rule selection. - Available functions include: \[ \cos, \sin, \tan, \cot, \sec, \csc \] - Available constants and symbols include: \[ \pi, (), \frac{}{} \]3. **Validation:** - After formulating the statement and selecting the rule, click the "Validate" button to check your work.---#### Proof Outline:To prove the identity, use the periodic properties of the tangent function, along with the identities for sine and cosine. Consider exploring:1. The periodicity of the tangent function.2. The angle subtraction formulas for sine and cosine.By systematically applying these rules, you can demonstrate that:\[ \tan(2\pi - x) = \frac{\sin(2\pi - x)}{\cos(2\pi - x)} = -\tan(x) \]This interactive proof exercise allows students to understand the underlying principles and visually explore the rules governing trigonometric identities. The image displays a series of options, each prefaced by a circular selection button, commonly known as a radio button. These options appear to be part of a multiple-choice quiz or selection form, likely centered on mathematical topics. The options listed are:1. Algebra2. Reciprocal3. Quotient4. Pythagorean5. Odd/EvenEach option has an adjacent circular button that can be selected, indicating this is likely part of a question where only one choice is allowed at a time. There are no graphs or diagrams present in the image.

Answered: Image /qna-images/answer/74c0fa33-233f-4d10-8fa4-ac0cb84c98df.jpg

Answered: Prove the identity. tan(27-x) = - tanx

Math

Trigonometry

Prove the identity. tan(27-x) = - tanx

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.2: Trigonometric Functions Of Angles

Problem 96E

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I need help proving the identity of the statement and the rule

### Proving the Trigonometric Identity

#### Objective:
Prove the following trigonometric identity:
\[ \tan(2\pi - x) = -\tan(x) \]

#### Instructions:
Each statement in your proof must be based on a Rule selected from the Rule menu. For additional details about a Rule, select the "More Information" button on the right of the Rule.

---

#### Steps:

1. **Statement and Rule Interface:**
- **Statement:**
\[ \tan(2\pi - x) \]
- **Rule:** Select an appropriate trigonometric rule from the drop-down menu.

2. **Interface Options:**
- The interface provides a list of trigonometric functions and constants for rule selection.
- Available functions include:
\[ \cos, \sin, \tan, \cot, \sec, \csc \]
- Available constants and symbols include:
\[ \pi, (), \frac{}{} \]

3. **Validation:**
- After formulating the statement and selecting the rule, click the "Validate" button to check your work.

---

#### Proof Outline:
To prove the identity, use the periodic properties of the tangent function, along with the identities for sine and cosine. Consider exploring:
1. The periodicity of the tangent function.
2. The angle subtraction formulas for sine and cosine.

By systematically applying these rules, you can demonstrate that:
\[ \tan(2\pi - x) = \frac{\sin(2\pi - x)}{\cos(2\pi - x)} = -\tan(x) \]

This interactive proof exercise allows students to understand the underlying principles and visually explore the rules governing trigonometric identities.

The image displays a series of options, each prefaced by a circular selection button, commonly known as a radio button. These options appear to be part of a multiple-choice quiz or selection form, likely centered on mathematical topics. The options listed are:

1. Algebra
2. Reciprocal
3. Quotient
4. Pythagorean
5. Odd/Even

Each option has an adjacent circular button that can be selected, indicating this is likely part of a question where only one choice is allowed at a time. There are no graphs or diagrams present in the image.

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