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: 13


$$\tan x(\cot x+\csc x)=1+\sec x$$

Work Step by Step

$$A=\tan x(\cot x+\csc x)$$ $$A=\tan x\cot x+\tan x\csc x$$ - Reciprocal Identity: $$\cot x=\frac{1}{\tan x}$$ Therefore, $$\tan x\cot x=1$$ Also, from Reciprocal Identity: $$\csc x=\frac{1}{\sin x}$$ and Quotient Identity: $$\tan x=\frac{\sin x}{\cos x}$$ We can make out that $$\tan x\csc x=\frac{\sin x}{\cos x}\times\frac{1}{\sin x}=\frac{1}{\cos x}=\sec x\hspace{1cm}\text{(Reciprocal Identity)}$$ Overall, we can now apply these results to $A$, $$A=1+\sec x$$
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