Trigonometry (10th Edition)

Published by Pearson
ISBN 10: 0321671775
ISBN 13: 978-0-32167-177-6

Chapter 5 - Trigonometric Identities - Section 5.1 Fundamental Identities - 5.1 Exercises - Page 195: 65

Answer

$$\sin^2\theta(\csc^2\theta-1)=\cos^2\theta$$

Work Step by Step

$$A=\sin^2\theta(\csc^2\theta-1)$$ - Pythagorean Identity: $$\cot^2\theta+1=\csc^2\theta$$ Replace into $A$: $$A=\sin^2\theta(\cot^2\theta+1-1)$$ $$A=\sin^2\theta(\cot^2\theta)$$ - Quotient Identity: $$\cot\theta=\frac{\cos\theta}{\sin\theta}$$ Replace into $A$: $$A=\sin^2\theta\times\frac{\cos^2\theta}{\sin^2\theta}$$ $$A=\cos^2\theta$$
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