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: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Fundamentals of Trigonometric Identities

Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.

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Chapter 1.4, Problem 56E

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The derivation of the identity sin2xcos2x=1−cos4x8

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Chapter 1.4, Problem 55E

Chapter 1.4, Problem 57E

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

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The derivation of the identity sin2xcos2x=1−cos4x8

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