Explanation Given: The identity tan x = sin 2 x 1 + cos 2 x 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 tan x = sin 2 x 1 + cos 2 x , let us take the right side of the equation and substitute for sin 2 x and cos 2 x using the double-angle formulas

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 1.4, Problem 55E

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The derivation of the identity tanx=sin2x1+cos2x

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Chapter 1.4, Problem 54E

Chapter 1.4, Problem 56E

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Chapter 1 Solutions

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