Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 5 - Trigonometric Identities - Section 5.4 Sum and Difference Identities for Sine and Tangent - 5.4 Exercises - Page 227: 45


$$\tan(2\pi-x)=-\tan x$$

Work Step by Step

$$X=\tan(2\pi-x)$$ According to tangent difference identity: $$\tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}$$ Expand $X$: $$X=\frac{\tan2\pi-\tan x}{1+\tan2\pi\tan x}$$ In the trigonometric circle, point $2\pi$ collides with point $0$. Therefore, $\tan2\pi=\tan0=0$. $$X=\frac{0-\tan x}{1+0\times\tan x}$$ $$X=\frac{-\tan x}{1}$$ $$X=-\tan x$$ Overall, $$\tan(2\pi-x)=-\tan x$$
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