Verify the identity. csc(x) – cot(x) sec(x) – 1 cot(x) sin(x) cos(x) sin(x) csc(x) – cot(x) sec(x) – 1 1 cos(x) sin(x) sin(x) cos(x) sin(x) cos(x) sin(x) 1 1 cos(x) - cos?(x) sin(x) – sin(x) cos(x) cono(1-(| cos(x sin(x)(1 - cos(x)) sin(x) = cot(x)
Verify the identity. csc(x) – cot(x) sec(x) – 1 cot(x) sin(x) cos(x) sin(x) csc(x) – cot(x) sec(x) – 1 1 cos(x) sin(x) sin(x) cos(x) sin(x) cos(x) sin(x) 1 1 cos(x) - cos?(x) sin(x) – sin(x) cos(x) cono(1-(| cos(x sin(x)(1 - cos(x)) sin(x) = cot(x)
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![## Verify the Identity
\[ \frac{\csc(x) - \cot(x)}{\sec(x) - 1} = \cot(x) \]
### Step-by-Step Solution
1. Start with the expression:
\[
\frac{\csc(x) - \cot(x)}{\sec(x) - 1}
\]
2. Rewrite using trigonometric identities:
\[
\frac{\frac{1}{\sin(x)} - \frac{\cos(x)}{\sin(x)}}{\frac{1}{\cos(x)} - 1}
\]
3. Simplify the expression:
\[
= \frac{\frac{1 - \cos(x)}{\sin(x)}}{\frac{1 - \cos(x)}{\cos(x)}}
\]
4. Multiply by the conjugate:
\[
= \frac{\frac{1 - \cos(x)}{\sin(x)}}{\frac{1 - \cos(x)}{\cos(x)}} \times \frac{\cos(x)}{\cos(x)}
\]
5. Further simplification leads to:
\[
= \frac{\sin(x) \cos(x)}{\sin(x)(1 - \cos(x))}
\]
6. Continue simplifying:
\[
= \frac{\cos(x)(1 - \cos(x))}{\sin(x)(1 - \cos(x))}
\]
7. Cancel common terms:
\[
= \frac{\cos(x)}{\sin(x)}
\]
8. Final answer is:
\[
= \cot(x)
\]
This completes the proof, and the original identity is verified.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90e413d6-0f95-45bc-9120-3de888a1b336%2F6869d694-cea6-46d0-b67c-9748f6347757%2Fn5cjnzl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Verify the Identity
\[ \frac{\csc(x) - \cot(x)}{\sec(x) - 1} = \cot(x) \]
### Step-by-Step Solution
1. Start with the expression:
\[
\frac{\csc(x) - \cot(x)}{\sec(x) - 1}
\]
2. Rewrite using trigonometric identities:
\[
\frac{\frac{1}{\sin(x)} - \frac{\cos(x)}{\sin(x)}}{\frac{1}{\cos(x)} - 1}
\]
3. Simplify the expression:
\[
= \frac{\frac{1 - \cos(x)}{\sin(x)}}{\frac{1 - \cos(x)}{\cos(x)}}
\]
4. Multiply by the conjugate:
\[
= \frac{\frac{1 - \cos(x)}{\sin(x)}}{\frac{1 - \cos(x)}{\cos(x)}} \times \frac{\cos(x)}{\cos(x)}
\]
5. Further simplification leads to:
\[
= \frac{\sin(x) \cos(x)}{\sin(x)(1 - \cos(x))}
\]
6. Continue simplifying:
\[
= \frac{\cos(x)(1 - \cos(x))}{\sin(x)(1 - \cos(x))}
\]
7. Cancel common terms:
\[
= \frac{\cos(x)}{\sin(x)}
\]
8. Final answer is:
\[
= \cot(x)
\]
This completes the proof, and the original identity is verified.
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