Explanation Given: The identity tan ( π − θ ) = − tan θ 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 Reflection identities 1. sin ( − x ) = − sin x 2. cos ( − x ) = cos x 3. tan ( − x ) = − tan x 4. sec ( − x ) = sec x 5. csc ( − x ) = − csc x 6. cot ( − x ) = − cot x From basic trigonometry we know that tan x = sin x cos x . We can substitute x = π − θ to get tan ( π − θ ) on the left side, and simplify the right side using addition formulas 1 and 2. Confirm that the right side simplifies to that of the given equation to prove the same. Calculations: Description To prove, tan ( π − θ ) = − tan θ Step 1: From basic trigonometry we know that tan x = sin x cos x

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 1.4, Problem 53E

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tan(π−θ)=−tanθ

The derivation of the identity

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Chapter 1.4, Problem 52E

Chapter 1.4, Problem 54E

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Chapter 1 Solutions

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tan ( π − θ ) = − tan θ The derivation of the identity

Solution Summary: The author explains that the given identity can be derived by using the basic trigonometric identities listed below.

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tan(π−θ)=−tanθ

The derivation of the identity

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Chapter 1 Solutions

tan ( π − θ ) = − tan θ The derivation of the identity

Solution Summary: The author explains that the given identity can be derived by using the basic trigonometric identities listed below.

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tan(π−θ)=−tanθ The derivation of the identity

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Chapter 1 Solutions

tan(π−θ)=−tanθ

The derivation of the identity