Use the fundamental identities to simplify the expression. (There is more than one correct form of the answer.) sin²(x) sec²(x) - sin²(x)
Use the fundamental identities to simplify the expression. (There is more than one correct form of the answer.) sin²(x) sec²(x) - sin²(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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![### Simplifying Trigonometric Expressions Using Identities
**Objective:**
Use the fundamental identities to simplify the given trigonometric expression. Note that there is more than one correct form of the answer.
**Expression to Simplify:**
\[ \sin^2(x) \sec^2(x) - \sin^2(x) \]
This educational exercise involves applying the foundational trigonometric identities to simplify the given expression.
**Step-by-Step Explanation:**
1. **Identify Relevant Identities:**
- \(\sec(x) = \frac{1}{\cos(x)}\)
- \(\sec^2(x) = 1 + \tan^2(x)\)
2. **Rewrite \(\sec^2(x)\) in terms of sine and cosine:**
\[ \sec^2(x) = 1 + \tan^2(x) \]
3. **Substitute the identity into the given expression:**
\[ \sin^2(x) \sec^2(x) - \sin^2(x) \]
\[ = \sin^2(x) (1 + \tan^2(x)) - \sin^2(x) \]
4. **Distribute \(\sin^2(x)\) across the terms within the parenthesis:**
\[ = \sin^2(x) + \sin^2(x) \tan^2(x) - \sin^2(x) \]
\[ = \sin^2(x) + \sin^2(x) \frac{\sin^2(x)}{\cos^2(x)} - \sin^2(x) \]
5. **Simplify the expression:**
\[ = \sin^2(x) + \frac{\sin^4(x)}{\cos^2(x)} - \sin^2(x) \]
\[ = \frac{\sin^4(x)}{\cos^2(x)} \]
\[ = \tan^2(x) \sin^2(x) \]
**Final Simplified Form(s):**
Thus, one of the simplified forms of the original expression is:
\[ \sin^2(x) \tan^2(x) \]
**Conclusion:**
This simplification process showcases the application of fundamental trigonometric identities to transform and simplify expressions. It demonstrates how recognizing and substituting known identities can help simplify complex trigonometric expressions](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0e38307-1ade-44bc-b712-aaeda4c58098%2F3751f066-754a-430c-8e21-0ff6de22a0e0%2Flrpncx_processed.png&w=3840&q=75)
Transcribed Image Text:### Simplifying Trigonometric Expressions Using Identities
**Objective:**
Use the fundamental identities to simplify the given trigonometric expression. Note that there is more than one correct form of the answer.
**Expression to Simplify:**
\[ \sin^2(x) \sec^2(x) - \sin^2(x) \]
This educational exercise involves applying the foundational trigonometric identities to simplify the given expression.
**Step-by-Step Explanation:**
1. **Identify Relevant Identities:**
- \(\sec(x) = \frac{1}{\cos(x)}\)
- \(\sec^2(x) = 1 + \tan^2(x)\)
2. **Rewrite \(\sec^2(x)\) in terms of sine and cosine:**
\[ \sec^2(x) = 1 + \tan^2(x) \]
3. **Substitute the identity into the given expression:**
\[ \sin^2(x) \sec^2(x) - \sin^2(x) \]
\[ = \sin^2(x) (1 + \tan^2(x)) - \sin^2(x) \]
4. **Distribute \(\sin^2(x)\) across the terms within the parenthesis:**
\[ = \sin^2(x) + \sin^2(x) \tan^2(x) - \sin^2(x) \]
\[ = \sin^2(x) + \sin^2(x) \frac{\sin^2(x)}{\cos^2(x)} - \sin^2(x) \]
5. **Simplify the expression:**
\[ = \sin^2(x) + \frac{\sin^4(x)}{\cos^2(x)} - \sin^2(x) \]
\[ = \frac{\sin^4(x)}{\cos^2(x)} \]
\[ = \tan^2(x) \sin^2(x) \]
**Final Simplified Form(s):**
Thus, one of the simplified forms of the original expression is:
\[ \sin^2(x) \tan^2(x) \]
**Conclusion:**
This simplification process showcases the application of fundamental trigonometric identities to transform and simplify expressions. It demonstrates how recognizing and substituting known identities can help simplify complex trigonometric expressions
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