Verify the identity. csc(x) - cot(x) - cot(x) sec(x) – 1 cos(x) sin(x) sin(x) csc(x) – cot(x) sec(x) – 1 1 cos(x) sin(x). sin(x) cos(x) sin(x) sin(x) cos(x) cos(x) - cos (x) sin(x) – sin(x) cos(x) cost»(1 - ([ sin(x)(1 – cos(x)) sin(x) cot(x) Need Help? Read It Watch It
Verify the identity. csc(x) - cot(x) - cot(x) sec(x) – 1 cos(x) sin(x) sin(x) csc(x) – cot(x) sec(x) – 1 1 cos(x) sin(x). sin(x) cos(x) sin(x) sin(x) cos(x) cos(x) - cos (x) sin(x) – sin(x) cos(x) cost»(1 - ([ sin(x)(1 – cos(x)) sin(x) cot(x) Need Help? Read It Watch It
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°.
Related questions
Question
Could you please write the answers clearly and understandably? Thank you.
![**Verifying the Trigonometric Identity**
The goal is to verify the trigonometric identity:
\[
\frac{\csc(x) - \cot(x)}{\sec(x) - 1} = \cot(x)
\]
**Step-by-Step Verification:**
1. **Start with LHS:**
\[
\frac{\csc(x) - \cot(x)}{\sec(x) - 1}
\]
2. **Reciprocal Trigonometric Identities:**
Using the identities \(\csc(x) = \frac{1}{\sin(x)}\), \(\cot(x) = \frac{\cos(x)}{\sin(x)}\), and \(\sec(x) = \frac{1}{\cos(x)}\), the expression becomes:
\[
\frac{\frac{1}{\sin(x)} - \frac{\cos(x)}{\sin(x)}}{\frac{1}{\cos(x)} - 1}
\]
3. **Simplify the Numerator:**
\[
= \frac{\frac{1 - \cos(x)}{\sin(x)}}{\frac{1}{\cos(x)} - 1}
\]
4. **Simplify the Denominator:**
\[
= \frac{1 - \cos(x)}{\sin(x)} \cdot \frac{\cos(x)}{1 - \cos(x)}
\]
5. **Cancel Common Terms:**
\[
= \frac{\cos(x)}{\sin(x)}
\]
6. **Simplified Result:**
\[
= \cot(x)
\]
**Conclusion:**
Thus, the identity is verified:
\[
\frac{\csc(x) - \cot(x)}{\sec(x) - 1} = \cot(x)
\]
Need Help?
- **Read It**: Access a detailed explanation and examples.
- **Watch It**: View a video tutorial for step-by-step guidance.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6658ba31-d576-43cc-a787-cd40c9f93584%2Fbd4d8c37-3349-46c0-a210-e623cfdaab98%2Fyup711_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Verifying the Trigonometric Identity**
The goal is to verify the trigonometric identity:
\[
\frac{\csc(x) - \cot(x)}{\sec(x) - 1} = \cot(x)
\]
**Step-by-Step Verification:**
1. **Start with LHS:**
\[
\frac{\csc(x) - \cot(x)}{\sec(x) - 1}
\]
2. **Reciprocal Trigonometric Identities:**
Using the identities \(\csc(x) = \frac{1}{\sin(x)}\), \(\cot(x) = \frac{\cos(x)}{\sin(x)}\), and \(\sec(x) = \frac{1}{\cos(x)}\), the expression becomes:
\[
\frac{\frac{1}{\sin(x)} - \frac{\cos(x)}{\sin(x)}}{\frac{1}{\cos(x)} - 1}
\]
3. **Simplify the Numerator:**
\[
= \frac{\frac{1 - \cos(x)}{\sin(x)}}{\frac{1}{\cos(x)} - 1}
\]
4. **Simplify the Denominator:**
\[
= \frac{1 - \cos(x)}{\sin(x)} \cdot \frac{\cos(x)}{1 - \cos(x)}
\]
5. **Cancel Common Terms:**
\[
= \frac{\cos(x)}{\sin(x)}
\]
6. **Simplified Result:**
\[
= \cot(x)
\]
**Conclusion:**
Thus, the identity is verified:
\[
\frac{\csc(x) - \cot(x)}{\sec(x) - 1} = \cot(x)
\]
Need Help?
- **Read It**: Access a detailed explanation and examples.
- **Watch It**: View a video tutorial for step-by-step guidance.
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