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4. Find the exact value of cos(83°) cos(23°) + sin(83°) sin(23⁰)

4. Find the exact value of cos(83°) cos(23°) + sin(83°) sin(23⁰)

Trigonometry (11th Edition)
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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Please answer number 4 and 5. Thank you!

## Trigonometric Identities and Formulas ### Problems: 1. Use basic identities to simplify the expression \( \sec x \cos x + \cos x - \frac{1}{\sec x} \). 2. Verify the identity \( \sin^2 x + \sec^2 x - 1 = \frac{(1 - \cos^2 x)(1 + \cos^2 x)}{\cos^2 x} \). 3. Find the exact value of \( \tan\left(\frac{7\pi}{12}\right) \) using the sum and difference formulas. 4. Find the exact value of \( \cos(83^\circ) \cos(23^\circ) + \sin(83^\circ) \sin(23^\circ) \). 5. Prove the identity \( \cos(4x) - \cos(3x) = \sin^2 x - 4\cos^2 x \sin^2 x \). These problems cover various fundamental aspects of trigonometric identities and their applications. Each question involves different techniques such as simplifying expressions, verifying identities, using sum and difference formulas, and proving more complex identities. ### Detailed Explanations: 1. **Simplifying Expressions Using Basic Trigonometric Identities**: - Use the definition of secant and the Pythagorean identities to simplify \( \sec x \cos x + \cos x - \frac{1}{\sec x} \). 2. **Verifying Trigonometric Identities**: - Demonstrate the equality between \( \sin^2 x + \sec^2 x - 1 \) and \( \frac{(1 - \cos^2 x)(1 + \cos^2 x)}{\cos^2 x} \) by manipulating the right-hand side expression using trigonometric identities. 3. **Using Sum and Difference Formulas**: - Calculate \( \tan\left(\frac{7\pi}{12}\right) \) by expressing it as the sum or difference of known angles where tangent values are easily accessible. 4. **Finding Exact Values Using Known Angles**: - Use known trigonometric values and identities to find the exact value for \( \cos(83^\circ) \cos(23^\circ) + \sin(83^\circ) \sin(23^\circ) \). 5. **
Transcribed Image Text:## Trigonometric Identities and Formulas ### Problems: 1. Use basic identities to simplify the expression \( \sec x \cos x + \cos x - \frac{1}{\sec x} \). 2. Verify the identity \( \sin^2 x + \sec^2 x - 1 = \frac{(1 - \cos^2 x)(1 + \cos^2 x)}{\cos^2 x} \). 3. Find the exact value of \( \tan\left(\frac{7\pi}{12}\right) \) using the sum and difference formulas. 4. Find the exact value of \( \cos(83^\circ) \cos(23^\circ) + \sin(83^\circ) \sin(23^\circ) \). 5. Prove the identity \( \cos(4x) - \cos(3x) = \sin^2 x - 4\cos^2 x \sin^2 x \). These problems cover various fundamental aspects of trigonometric identities and their applications. Each question involves different techniques such as simplifying expressions, verifying identities, using sum and difference formulas, and proving more complex identities. ### Detailed Explanations: 1. **Simplifying Expressions Using Basic Trigonometric Identities**: - Use the definition of secant and the Pythagorean identities to simplify \( \sec x \cos x + \cos x - \frac{1}{\sec x} \). 2. **Verifying Trigonometric Identities**: - Demonstrate the equality between \( \sin^2 x + \sec^2 x - 1 \) and \( \frac{(1 - \cos^2 x)(1 + \cos^2 x)}{\cos^2 x} \) by manipulating the right-hand side expression using trigonometric identities. 3. **Using Sum and Difference Formulas**: - Calculate \( \tan\left(\frac{7\pi}{12}\right) \) by expressing it as the sum or difference of known angles where tangent values are easily accessible. 4. **Finding Exact Values Using Known Angles**: - Use known trigonometric values and identities to find the exact value for \( \cos(83^\circ) \cos(23^\circ) + \sin(83^\circ) \sin(23^\circ) \). 5. **
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