cos cos² sec
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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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
Prove the following statement is an identity by transforming the left side of the equation to the right side.
![### Mathematical Expression Breakdown
The image contains two trigonometric expressions.
1. **Expression 1:**
\[\frac{\cos{\theta}}{\sec{\theta}}\]
2. **Expression 2:**
\[\cos^2{\theta}\]
### Detailed Description:
1. **Expression 1 Explanation:**
The first expression is the quotient of cosine (cos) of an angle \(\theta\) and secant (sec) of the same angle. According to trigonometric identities, secant is the reciprocal of cosine:
\[ \sec{\theta} = \frac{1}{\cos{\theta}} \]
Thus, the expression simplifies as follows:
\[ \frac{\cos{\theta}}{\sec{\theta}} = \frac{\cos{\theta}}{\frac{1}{\cos{\theta}}} = \cos^2{\theta} \]
2. **Expression 2 Explanation:**
The second expression represents the square of the cosine of angle \(\theta\):
\[\cos^2{\theta} = (\cos{\theta})^2\]
### Conclusion:
Both expressions, \(\frac{\cos{\theta}}{\sec{\theta}}\) and \(\cos^2{\theta}\), are equivalent and simplify to the same value. This showcases a fundamental relationship in trigonometry between cosine and secant functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F242c3500-5eba-4bd1-9201-b2fb9263f121%2F3bb69b10-ff57-43f0-ab80-c2b8ce9949bf%2Fn806hns.jpeg&w=3840&q=75)
Transcribed Image Text:### Mathematical Expression Breakdown
The image contains two trigonometric expressions.
1. **Expression 1:**
\[\frac{\cos{\theta}}{\sec{\theta}}\]
2. **Expression 2:**
\[\cos^2{\theta}\]
### Detailed Description:
1. **Expression 1 Explanation:**
The first expression is the quotient of cosine (cos) of an angle \(\theta\) and secant (sec) of the same angle. According to trigonometric identities, secant is the reciprocal of cosine:
\[ \sec{\theta} = \frac{1}{\cos{\theta}} \]
Thus, the expression simplifies as follows:
\[ \frac{\cos{\theta}}{\sec{\theta}} = \frac{\cos{\theta}}{\frac{1}{\cos{\theta}}} = \cos^2{\theta} \]
2. **Expression 2 Explanation:**
The second expression represents the square of the cosine of angle \(\theta\):
\[\cos^2{\theta} = (\cos{\theta})^2\]
### Conclusion:
Both expressions, \(\frac{\cos{\theta}}{\sec{\theta}}\) and \(\cos^2{\theta}\), are equivalent and simplify to the same value. This showcases a fundamental relationship in trigonometry between cosine and secant functions.
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