To prove the trigonometric identity, follow the steps below by selecting the appropriate rule justifying each transformation: Given Identity: \[ \sin^2 x - \sin^4 x = \cos^4 x - \cos^2 x \] Steps and Statements: 1. Start with: \[ \sin^2 x - \sin^4 x \] 2. Factor out $\sin^2 x$: \[ \sin^2 x (1 - \sin^2 x) \quad \text{(use Rule 1)} \] 3. Use the Pythagorean identity $1 - \sin^2 x = \cos^2 x$: \[ \cos^2 x \cdot \sin^2 x \quad \text{(use Rule 2)} \] 4. Use the identity again to substitute: \[ \cos^2 x \cdot (1 - \cos^2 x) \quad \text{(use Rule 2)} \] 5. Finally, express as: \[ \cos^4 x - \cos^2 x \] This demonstrates the equality of both sides of the given identity using fundamental trigonometric identities.

To prove the trigonometric identity, follow the steps below by selecting the appropriate rule justifying each transformation: Given Identity: \[ \sin^2 x - \sin^4 x = \cos^4 x - \cos^2 x \] Steps and Statements: 1. Start with: \[ \sin^2 x - \sin^4 x \] 2. Factor out $\sin^2 x$: \[ \sin^2 x (1 - \sin^2 x) \quad \text{(use Rule 1)} \] 3. Use the Pythagorean identity $1 - \sin^2 x = \cos^2 x$: \[ \cos^2 x \cdot \sin^2 x \quad \text{(use Rule 2)} \] 4. Use the identity again to substitute: \[ \cos^2 x \cdot (1 - \cos^2 x) \quad \text{(use Rule 2)} \] 5. Finally, express as: \[ \cos^4 x - \cos^2 x \] This demonstrates the equality of both sides of the given identity using fundamental trigonometric identities.

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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Question

To prove the trigonometric identity, follow the steps below by selecting the appropriate rule justifying each transformation:

Given Identity:
\[
\sin^2 x - \sin^4 x = \cos^4 x - \cos^2 x
\]

Steps and Statements:

1. Start with:
\[
\sin^2 x - \sin^4 x
\]

2. Factor out $\sin^2 x$:
\[
\sin^2 x (1 - \sin^2 x) \quad \text{(use Rule 1)}
\]

3. Use the Pythagorean identity $1 - \sin^2 x = \cos^2 x$:
\[
\cos^2 x \cdot \sin^2 x \quad \text{(use Rule 2)}
\]

4. Use the identity again to substitute:
\[
\cos^2 x \cdot (1 - \cos^2 x) \quad \text{(use Rule 2)}
\]

5. Finally, express as:
\[
\cos^4 x - \cos^2 x
\]

This demonstrates the equality of both sides of the given identity using fundamental trigonometric identities.

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