Factor the expression. Use the fundamental identities to simplify, if necessary. (There is more than one correct form of each answer.) cos(x) - 2 cos²(x) - 4
Factor the expression. Use the fundamental identities to simplify, if necessary. (There is more than one correct form of each answer.) cos(x) - 2 cos²(x) - 4
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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![**Problem Statement:**
Factor the expression. Use the fundamental identities to simplify, if necessary. (There is more than one correct form of each answer.)
\[
\frac{\cos(x) - 2}{\cos^2(x) - 4}
\]
---
**Solution:**
To factor and simplify the given trigonometric expression, follow these steps:
1. **Identify the Denominator as a Difference of Squares:**
\[\cos^2(x) - 4\]
This can be factored as:
\[(\cos(x))^2 - 2^2\]
\[\cos(x) - 2\cos(x) + 2\]
Therefore:
\[\cos^2(x) - 4 = (\cos(x) - 2)(\cos(x) + 2)\]
2. **Factor the Expression:**
Using the factored form of the denominator:
\[\frac{\cos(x) - 2}{(\cos(x) - 2)(\cos(x) + 2)}\]
3. **Simplify the Expression:**
Assuming \(\cos(x) \neq 2\) (to avoid division by zero):
\[\frac{\cos(x) - 2}{(\cos(x) - 2)(\cos(x) + 2)} = \frac{1}{\cos(x) + 2}\]
Therefore, the simplified form is:
\[\boxed{\frac{1}{\cos(x) + 2}}\]
There might be other equivalent forms depending on the application of further trigonometric identities.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0e38307-1ade-44bc-b712-aaeda4c58098%2F3c92072e-596e-4084-811f-96a91760c6de%2Fu2a38ui_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Factor the expression. Use the fundamental identities to simplify, if necessary. (There is more than one correct form of each answer.)
\[
\frac{\cos(x) - 2}{\cos^2(x) - 4}
\]
---
**Solution:**
To factor and simplify the given trigonometric expression, follow these steps:
1. **Identify the Denominator as a Difference of Squares:**
\[\cos^2(x) - 4\]
This can be factored as:
\[(\cos(x))^2 - 2^2\]
\[\cos(x) - 2\cos(x) + 2\]
Therefore:
\[\cos^2(x) - 4 = (\cos(x) - 2)(\cos(x) + 2)\]
2. **Factor the Expression:**
Using the factored form of the denominator:
\[\frac{\cos(x) - 2}{(\cos(x) - 2)(\cos(x) + 2)}\]
3. **Simplify the Expression:**
Assuming \(\cos(x) \neq 2\) (to avoid division by zero):
\[\frac{\cos(x) - 2}{(\cos(x) - 2)(\cos(x) + 2)} = \frac{1}{\cos(x) + 2}\]
Therefore, the simplified form is:
\[\boxed{\frac{1}{\cos(x) + 2}}\]
There might be other equivalent forms depending on the application of further trigonometric identities.
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