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: 41


$$\frac{\sin^2x}{\cos^2x}+\sin x\csc x=\sec^2x$$

Work Step by Step

$$A=\frac{\sin^2x}{\cos^2x}+\sin x\csc x$$ $$A=\Big(\frac{\sin x}{\cos x}\Big)^2+\sin x\csc x$$ - Quotient Identity: $$\frac{\sin x}{\cos x}=\tan x$$ - Reciprocal Identity: $$\csc x=\frac{1}{\sin x}$$ Replace them into $A$, we have $$A=\tan^2 x+\sin x\times\frac{1}{\sin x}$$ $$A=\tan^2 x+1$$ - Pythagorean Identity: $$\tan^2 x+1=\sec^2 x$$ which means $$A=\sec^2x$$
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