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 5.2 Verifying Trigonometric Identities - 5.2 Exercises - Page 208: 18


$$(\tan x+\cot x)^2=\sec^2 x+\csc^2 x$$

Work Step by Step

$$A=(\tan x+\cot x)^2$$ $$A=\tan^2x+2\tan x\cot x+\cot^2 x$$ $$A=\tan^2 x+\cot^2 x+2\tan x\cot x$$ - According to a Reciprocal Identity: $$\cot x=\frac{1}{\tan x}$$ which means $$\cot x\tan x=1$$ Therefore, $A$ would be $$A=\tan^2 x+\cot^2 x+2\times1$$ $$A=\tan^2 x+\cot^2 x+2$$ $$A=(\tan^2 x+1)+(\cot^2 x+1)$$ - We have 2 following Pythagorean Identities: $$\tan^2 x+1=\sec^2 x$$ and $$\cot^2 x+1=\csc^2 x$$ Apply them to $A$: $$A=\sec^2 x+\csc^2 x$$
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