**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.

Answered: Image /qna-images/answer/b905e43f-5b93-44b6-884c-21c9fd1a2268.jpg

Answered: Verify the identity. (1 - sin?(t) + 3…

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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Question

**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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