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) +

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Chapter1: Functions And Models
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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.
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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