**Trigonometric Identity Verification****Objective:**Verify the identity by converting the left side into sines and cosines. (Simplify at each step.)**Given Identity:**\[ \cot(x) - \tan(x) = \sec(x)(\csc(x) - 2 \sin(x)) \]### Step-by-Step Verification:1. **Express \(\cot(x)\) and \(\tan(x)\) in terms of sine and cosine:** \[\cot(x) - \tan(x) = \frac{\cos(x)}{\sin(x)} - \frac{\sin(x)}{\cos(x)}\]2. **Find a common denominator:** \[ \frac{\cos^2(x) - \sin^2(x)}{\sin(x)\cos(x)} \]3. **Simplify using the Pythagorean identity \(\cos^2(x) + \sin^2(x) = 1\):** \[ \cos^2(x) - \sin^2(x) = \cos(x) (\cos(x)) - \sin^2(x) \] Substitution leads to: \[ = \sin(x) \left( \cos(x) \right) \] \[ = \frac{1 - (\sin^2(x) + \sin^2(x))}{\sin(x) \cos(x)} \] This simplifies to: \[ \frac{1 - 2\sin^2(x)}{\sin(x) \cos(x)} \]4. **Separate the expression into:** \[ \frac{1}{\cos(x)} \cdot \frac{1 - 2 \sin^2(x)}{\sin(x)} \]5. **Break it down further:** \[ \sec(x) \left( \frac{1 - 2\sin^2(x)}{\sin(x)} \right) \]6. **Notice the resemblance to the given identity:** \[ \sec(x) \left( \csc(x) - 2 \sin(x) \right) \]### Verification Complete:\[\cot(x) - \tan(x) = \sec(x) \left( \csc(x) - 2 \sin(x) \right)\]### Explanation of Incorrect Steps:- In

Topic: Trigonometry The answer is given below

Verify the identity by converting the left side into sines and cosines. (Simplify at each step.) cot(x) =tan(x) = sec(x)(csc(x) - 2 sin(x)) cos(x) _ _sin(x) sin(x) cos(x) cos²(x) — sin²(x) cot(x) - tan(x) = = 11 = = sin(x) cos(x) − (sin²(x) + sin²(x)) cos(x) 1 - 2 sin²(x) 1 cos(x) X 1 - 2 sin²(x) 2 sin²(x) 1 1 cos(x) sin(x) sec(x) (csc(x) — 2 sin(x)) X

Verify the identity by converting the left side into sines and cosines. (Simplify at each step.) cot(x) =tan(x) = sec(x)(csc(x) - 2 sin(x)) cos(x) _ _sin(x) sin(x) cos(x) cos²(x) — sin²(x) cot(x) - tan(x) = = 11 = = sin(x) cos(x) − (sin²(x) + sin²(x)) cos(x) 1 - 2 sin²(x) 1 cos(x) X 1 - 2 sin²(x) 2 sin²(x) 1 1 cos(x) sin(x) sec(x) (csc(x) — 2 sin(x)) X

Trigonometry (11th Edition)

11th Edition

ISBN:9780134217437

Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Chapter1: Trigonometric Functions

Section: Chapter Questions

Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.

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Question

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**Trigonometric Identity Verification**

**Objective:**
Verify the identity by converting the left side into sines and cosines. (Simplify at each step.)

**Given Identity:**
\[ \cot(x) - \tan(x) = \sec(x)(\csc(x) - 2 \sin(x)) \]

### Step-by-Step Verification:

1. **Express \(\cot(x)\) and \(\tan(x)\) in terms of sine and cosine:**
\[\cot(x) - \tan(x) = \frac{\cos(x)}{\sin(x)} - \frac{\sin(x)}{\cos(x)}\]

2. **Find a common denominator:**
\[ \frac{\cos^2(x) - \sin^2(x)}{\sin(x)\cos(x)} \]

3. **Simplify using the Pythagorean identity \(\cos^2(x) + \sin^2(x) = 1\):**

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

Substitution leads to:
\[ = \sin(x) \left( \cos(x) \right) \]

\[ = \frac{1 - (\sin^2(x) + \sin^2(x))}{\sin(x) \cos(x)} \]

This simplifies to:
\[ \frac{1 - 2\sin^2(x)}{\sin(x) \cos(x)} \]

4. **Separate the expression into:**
\[ \frac{1}{\cos(x)} \cdot \frac{1 - 2 \sin^2(x)}{\sin(x)} \]

5. **Break it down further:**
\[ \sec(x) \left( \frac{1 - 2\sin^2(x)}{\sin(x)} \right) \]

6. **Notice the resemblance to the given identity:**
\[ \sec(x) \left( \csc(x) - 2 \sin(x) \right) \]

### Verification Complete:
\[
\cot(x) - \tan(x) = \sec(x) \left( \csc(x) - 2 \sin(x) \right)
\]

### Explanation of Incorrect Steps:
- In

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