Prove the identity. secx- tanx = 1 sec 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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I have solved one of the steps, how can I solve the rest? ALEKS wants to me to list each step and the rule.
Transcribed Image Text:The image displays a multiple-choice selection list with five mathematics-related options. The options are as follows:
1. Algebra (Radio button: unselected)
2. Reciprocal (Radio button: unselected)
3. Quotient (Radio button: unselected)
4. Pythagorean (Radio button: unselected)
5. Odd/Even (Radio button: unselected)
![### Proving the Trigonometric Identity
#### Prove the identity:
\[ \sec^2{x} - \tan^2{x} = 1 \]
Note that each statement must be based on a Rule chosen from the Rule menu. To see a detailed description of a Rule, select the More Information Button to the right of the Rule.
---
#### Statements and Rules
| **Statement** | **Rule** |
|-----------------------------------|-----------|
| \(\sec^2{x} - \tan^2{x} = 1\) | Algebra |
To understand the proof of the given identity, let's break it down step-by-step using the fundamental trigonometric identities.
---
#### Detailed Explanation
##### Trigonometric Identities Used:
1. \(\sec{x} = \frac{1}{\cos{x}}\)
2. \(\tan{x} = \frac{\sin{x}}{\cos{x}}\)
3. Pythagorean Identity: \(\sin^2{x} + \cos^2{x} = 1\)
Given the identity to prove:
\[ \sec^2{x} - \tan^2{x} = 1 \]
By expressing \(\sec{x}\) and \(\tan{x}\) using their definitions in terms of sine and cosine:
\[ \sec^2{x} = \left(\frac{1}{\cos{x}}\right)^2 = \frac{1}{\cos^2{x}} \]
\[ \tan^2{x} = \left(\frac{\sin{x}}{\cos{x}}\right)^2 = \frac{\sin^2{x}}{\cos^2{x}} \]
Substitute these into the original equation:
\[ \frac{1}{\cos^2{x}} - \frac{\sin^2{x}}{\cos^2{x}} \]
Combine the fractions:
\[ \frac{1 - \sin^2{x}}{\cos^2{x}} \]
Using the Pythagorean identity:
\[ 1 - \sin^2{x} = \cos^2{x} \]
Substitute this back into the equation:
\[ \frac{\cos^2{x}}{\cos^2{x}} = 1 \]
Hence, the identity is proven:
\[ \sec^2{x} - \tan^2{x} = 1 \]
---
#### Interactive Diagram
To visualize how rules relate to different](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F38188fd7-32af-4cc1-9ec8-e56982db8708%2F7c5d9227-fea0-49ca-80b0-2536f727b685%2F4u0sgwl.png&w=3840&q=75)
Transcribed Image Text:### Proving the Trigonometric Identity
#### Prove the identity:
\[ \sec^2{x} - \tan^2{x} = 1 \]
Note that each statement must be based on a Rule chosen from the Rule menu. To see a detailed description of a Rule, select the More Information Button to the right of the Rule.
---
#### Statements and Rules
| **Statement** | **Rule** |
|-----------------------------------|-----------|
| \(\sec^2{x} - \tan^2{x} = 1\) | Algebra |
To understand the proof of the given identity, let's break it down step-by-step using the fundamental trigonometric identities.
---
#### Detailed Explanation
##### Trigonometric Identities Used:
1. \(\sec{x} = \frac{1}{\cos{x}}\)
2. \(\tan{x} = \frac{\sin{x}}{\cos{x}}\)
3. Pythagorean Identity: \(\sin^2{x} + \cos^2{x} = 1\)
Given the identity to prove:
\[ \sec^2{x} - \tan^2{x} = 1 \]
By expressing \(\sec{x}\) and \(\tan{x}\) using their definitions in terms of sine and cosine:
\[ \sec^2{x} = \left(\frac{1}{\cos{x}}\right)^2 = \frac{1}{\cos^2{x}} \]
\[ \tan^2{x} = \left(\frac{\sin{x}}{\cos{x}}\right)^2 = \frac{\sin^2{x}}{\cos^2{x}} \]
Substitute these into the original equation:
\[ \frac{1}{\cos^2{x}} - \frac{\sin^2{x}}{\cos^2{x}} \]
Combine the fractions:
\[ \frac{1 - \sin^2{x}}{\cos^2{x}} \]
Using the Pythagorean identity:
\[ 1 - \sin^2{x} = \cos^2{x} \]
Substitute this back into the equation:
\[ \frac{\cos^2{x}}{\cos^2{x}} = 1 \]
Hence, the identity is proven:
\[ \sec^2{x} - \tan^2{x} = 1 \]
---
#### Interactive Diagram
To visualize how rules relate to different
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