Verify the following identities. Be sure to show all steps in a clear and neat sequence. 1. sin x + coS x cot x = CSC X
Verify the following identities. Be sure to show all steps in a clear and neat sequence. 1. sin x + coS x cot x = CSC X
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°.
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Question
Verify the identities
Transcribed Image Text:Verify the following identities. Be sure to show **all** steps in a clear and neat sequence.
1. \( \sin x + \cos x \cot x = \csc x \)
![The image contains a trigonometric identity problem labeled as question 2. The equation is:
\[ \frac{\sin t}{1 - \cos t} = \csc t + \cot t \]
This equation involves trigonometric functions: sine (\(\sin\)), cosine (\(\cos\)), cosecant (\(\csc\)), and cotangent (\(\cot\)). The goal is typically to prove or verify that the left side of the equation is equivalent to the right side using known trigonometric identities.
### Explanation:
- **Left Side**: The expression \(\frac{\sin t}{1 - \cos t}\) requires manipulation or identities to simplify.
- **Right Side**: The expression \(\csc t + \cot t\) can be rewritten in terms of sine and cosine:
\[
\csc t = \frac{1}{\sin t}, \quad \cot t = \frac{\cos t}{\sin t}
\]
Therefore:
\[
\csc t + \cot t = \frac{1}{\sin t} + \frac{\cos t}{\sin t} = \frac{1 + \cos t}{\sin t}
\]
Students should use trigonometric identities to manipulate both sides until they match, such as Pythagorean identities or algebraic manipulation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa831237a-6047-4ff6-af83-c454e7106310%2F1f3bb544-221a-4370-aa73-5424581950c6%2F0p6ttiq.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains a trigonometric identity problem labeled as question 2. The equation is:
\[ \frac{\sin t}{1 - \cos t} = \csc t + \cot t \]
This equation involves trigonometric functions: sine (\(\sin\)), cosine (\(\cos\)), cosecant (\(\csc\)), and cotangent (\(\cot\)). The goal is typically to prove or verify that the left side of the equation is equivalent to the right side using known trigonometric identities.
### Explanation:
- **Left Side**: The expression \(\frac{\sin t}{1 - \cos t}\) requires manipulation or identities to simplify.
- **Right Side**: The expression \(\csc t + \cot t\) can be rewritten in terms of sine and cosine:
\[
\csc t = \frac{1}{\sin t}, \quad \cot t = \frac{\cos t}{\sin t}
\]
Therefore:
\[
\csc t + \cot t = \frac{1}{\sin t} + \frac{\cos t}{\sin t} = \frac{1 + \cos t}{\sin t}
\]
Students should use trigonometric identities to manipulate both sides until they match, such as Pythagorean identities or algebraic manipulation.
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