Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Appendix D - Trigonometry - D Exercises: 48

Answer

$tan^{2}\alpha-sin^{2}\alpha=tan^{2}\alpha$ $sin^{2}\alpha$

Work Step by Step

Need to prove the identity $tan^{2}\alpha-sin^{2}\alpha=tan^{2}\alpha$ $sin^{2}\alpha$ In order to prove this, take left side isolate. $tan^{2}\alpha-sin^{2}\alpha= \frac{sin^{2}\alpha}{cos^{2}\alpha}-sin^{2}\alpha$ $tan^{2}\alpha-sin^{2}\alpha=\frac{sin^{2}\alpha-sin^{2}\alpha cos^{2}\alpha}{cos^{2}\alpha}$ Thus, $tan^{2}\alpha-sin^{2}\alpha=\frac{1-cos^{2}}{cos^{2}\alpha}\times sin^{2}\alpha$ Since, ${1-cos^{2}\alpha}=sin^{2}\alpha$, Now, $tan^{2}\alpha-sin^{2}\alpha=\frac{sin^{2}\alpha}{cos^{2}\alpha}\times sin^{2}\alpha$ Hence,$tan^{2}\alpha-sin^{2}\alpha=tan^{2}\alpha$ $sin^{2}\alpha$
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