Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Question
## 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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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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