Use the fundamental identities to simplify the expression. (There is more than one correct form of the answer.) tan(8) cot(0) sec(8)
Use the fundamental identities to simplify the expression. (There is more than one correct form of the answer.) tan(8) cot(0) sec(8)
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
![### Simplifying Trigonometric Expressions Using Fundamental Identities
#### Problem Statement:
Use the fundamental identities to simplify the expression. (There is more than one correct form of the answer.)
\[
\frac{\tan(\theta) \cot(\theta)}{\sec(\theta)}
\]
#### Instructions:
1. Recognize and apply fundamental trigonometric identities to simplify the given expression.
2. Note that multiple simplified forms might be valid as long as they adhere to trigonometric identities.
In the given expression:
- \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)
- \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\)
- \(\sec(\theta) = \frac{1}{\cos(\theta)}\)
Starting with \(\tan(\theta) \cot(\theta)\):
\[
\tan(\theta) \cot(\theta) = \left(\frac{\sin(\theta)}{\cos(\theta)}\right) \left(\frac{\cos(\theta)}{\sin(\theta)}\right)
= \frac{\sin(\theta) \cos(\theta)}{\cos(\theta) \sin(\theta)}
= 1
\]
Therefore, the original expression simplifies to:
\[
\frac{\tan(\theta) \cot(\theta)}{\sec(\theta)} = \frac{1}{\sec(\theta)} = \cos(\theta)
\]
Thus, a simplified form of the given expression is:
\[
\cos(\theta)
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0e38307-1ade-44bc-b712-aaeda4c58098%2Fa9faafde-e520-4ace-a8d2-4ffceb537e6a%2Fexlra98_processed.png&w=3840&q=75)
Transcribed Image Text:### Simplifying Trigonometric Expressions Using Fundamental Identities
#### Problem Statement:
Use the fundamental identities to simplify the expression. (There is more than one correct form of the answer.)
\[
\frac{\tan(\theta) \cot(\theta)}{\sec(\theta)}
\]
#### Instructions:
1. Recognize and apply fundamental trigonometric identities to simplify the given expression.
2. Note that multiple simplified forms might be valid as long as they adhere to trigonometric identities.
In the given expression:
- \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)
- \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\)
- \(\sec(\theta) = \frac{1}{\cos(\theta)}\)
Starting with \(\tan(\theta) \cot(\theta)\):
\[
\tan(\theta) \cot(\theta) = \left(\frac{\sin(\theta)}{\cos(\theta)}\right) \left(\frac{\cos(\theta)}{\sin(\theta)}\right)
= \frac{\sin(\theta) \cos(\theta)}{\cos(\theta) \sin(\theta)}
= 1
\]
Therefore, the original expression simplifies to:
\[
\frac{\tan(\theta) \cot(\theta)}{\sec(\theta)} = \frac{1}{\sec(\theta)} = \cos(\theta)
\]
Thus, a simplified form of the given expression is:
\[
\cos(\theta)
\]
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