Explanation Given: The identity sin 2 x cos 2 x = 1 − cos 4 x 8 The given identity can be derived by using the identities listed below. Basic trigonometric identities : Pythagorean identity sin 2 x + cos 2 x = 1 Complementary angles 1. sin ( π 2 − x ) = cos x 2. cos ( π 2 − x ) = sin x Addition formulas 1. sin ( x + y ) = sin x cos y + cos x sin y 2. cos ( x + y ) = cos x cos y − sin x sin y Double-angle formulas 1. sin 2 x = 1 2 ( 1 − cos 2 x ) 2. cos 2 x = 1 2 ( 1 + cos 2 x ) 3. cos 2 x = cos 2 x − sin 2 x 4. sin 2 x = 2 sin x cos x Shift formulas 1. sin ( x + π 2 ) = cos x 2. cos ( x + π 2 ) = − sin x To prove sin 2 x cos 2 x = 1 − cos 4 x 8 , let us take the left side of the equation and substitute for sin 2 x and cos 2 x using the double-angle formulas. Confirm that it simplifies to the right side of the equation. Formula used: Algebraic identity ( a + b ) ( a − b ) = a 2 − b 2 Calculations: Description To prove, sin 2 x cos 2 x = 1 − cos 4 x 8 Step 1: let us take the left side of the equation...

3rd Edition

ISBN: 9781464114885

Author: Jon Rogawski, Colin Adams

Publisher: W. H. Freeman

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Question

Chapter 1.4, Problem 56E

To determine

The derivation of the identity $\sin^{2} x \cos^{2} x = \frac{1 - \cos 4 x}{8}$

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Chapter 1.4, Problem 55E

Chapter 1.4, Problem 57E

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Chapter 1 Solutions

Calculus: Early Transcendentals

Ch. 1.1 - Prob. 1PQ Ch. 1.1 - Prob. 2PQ Ch. 1.1 - Prob. 3PQ Ch. 1.1 - Prob. 4PQ Ch. 1.1 - Prob. 5PQ Ch. 1.1 - Prob. 6PQ Ch. 1.1 - Prob. 7PQ Ch. 1.1 - Prob. 8PQ Ch. 1.1 - Prob. 9PQ Ch. 1.1 - Prob. 10PQ

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Knowledge Booster

The derivation of the identity sin 2 x cos 2 x = 1 − cos 4 x 8

The derivation of the identity sin2xcos2x=1−cos4x8

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Chapter 1 Solutions

The derivation of the identity $\sin^{2} x \cos^{2} x = \frac{1 - \cos 4 x}{8}$