Verify the identity. (1 - sin?(t) + 3 cos²(t))2 + 16 sin?(t) cos?(t) = 16 cos?(t) (1 - sin?(t) + 3 cos?(t))2 + 16 sin2(t) cos?(t) (4 cos?(t))2 + = 16 cos?(t)(cos?(t) +
Verify the identity. (1 - sin?(t) + 3 cos²(t))2 + 16 sin?(t) cos?(t) = 16 cos?(t) (1 - sin?(t) + 3 cos?(t))2 + 16 sin2(t) cos?(t) (4 cos?(t))2 + = 16 cos?(t)(cos?(t) +
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Verify the identity.**
\[
(1 - \sin^2(t) + 3 \cos^2(t))^2 + 16 \sin^2(t) \cos^2(t) = 16 \cos^2(t)
\]
\[
(1 - \sin^2(t) + 3 \cos^2(t))^2 + 16 \sin^2(t) \cos^2(t) = (4 \cos^2(t))^2 + \boxed{\phantom{1}}
\]
\[
= \ 16 \cos^2(t)(\cos^2(t) + \boxed{\phantom{1}})
\]
# Explanation:
This transcription deals with a trigonometric identity verification problem. The equation involves verifying that two expressions involving trigonometric functions are equivalent.
1. **Original Equation**:
- The left-hand side consists of two parts: the square of a binomial expression and a product involving \(\sin^2(t)\) and \(\cos^2(t)\).
- The right-hand side is a simple multiple of \(\cos^2(t)\).
2. **Breaking Down the Identity**:
- The given identity involves manipulating trigonometric identities for sine and cosine, particularly leveraging the Pythagorean identity \( \sin^2(t) + \cos^2(t) = 1 \).
3. **Steps to Solution**:
- The identity that needs verifying begins with expanding and simplifying the left-hand side to reach an equivalent expression in terms of \(\cos^2(t)\).
- The boxed spaces imply missing steps or intermediate simplification efforts. Filling these in would involve algebraic manipulations showing equivalence to the right-hand side.
This exercise strengthens the understanding of manipulating and transforming trigonometric identities.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F69378f5f-597f-4bf2-b9b0-d667578bc76e%2Fb905e43f-5b93-44b6-884c-21c9fd1a2268%2Fl3h169c_processed.png&w=3840&q=75)
Transcribed Image Text:**Verify the identity.**
\[
(1 - \sin^2(t) + 3 \cos^2(t))^2 + 16 \sin^2(t) \cos^2(t) = 16 \cos^2(t)
\]
\[
(1 - \sin^2(t) + 3 \cos^2(t))^2 + 16 \sin^2(t) \cos^2(t) = (4 \cos^2(t))^2 + \boxed{\phantom{1}}
\]
\[
= \ 16 \cos^2(t)(\cos^2(t) + \boxed{\phantom{1}})
\]
# Explanation:
This transcription deals with a trigonometric identity verification problem. The equation involves verifying that two expressions involving trigonometric functions are equivalent.
1. **Original Equation**:
- The left-hand side consists of two parts: the square of a binomial expression and a product involving \(\sin^2(t)\) and \(\cos^2(t)\).
- The right-hand side is a simple multiple of \(\cos^2(t)\).
2. **Breaking Down the Identity**:
- The given identity involves manipulating trigonometric identities for sine and cosine, particularly leveraging the Pythagorean identity \( \sin^2(t) + \cos^2(t) = 1 \).
3. **Steps to Solution**:
- The identity that needs verifying begins with expanding and simplifying the left-hand side to reach an equivalent expression in terms of \(\cos^2(t)\).
- The boxed spaces imply missing steps or intermediate simplification efforts. Filling these in would involve algebraic manipulations showing equivalence to the right-hand side.
This exercise strengthens the understanding of manipulating and transforming trigonometric identities.
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