Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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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](https://content.bartleby.com/qna-images/question/a0e38307-1ade-44bc-b712-aaeda4c58098/d8e72f42-34fc-43e2-91aa-36134763250f/wp1zwg_processed.png)
Transcribed Image Text:**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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