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: 11


$$\frac{1}{\sin\theta\cos\theta}$$ $$or$$ $$\csc\theta\sec\theta$$

Work Step by Step

$$\cot\theta+\frac{1}{\cot\theta}$$ Multiply $\cot\theta$ by $\frac{\cot\theta}{\cot\theta}$ to get common denominators: $$\frac{\cot^{2}\theta}{\cot\theta}+\frac{1}{\cot\theta}$$ $$=\frac{\cot^{2}\theta+1}{\cot\theta}$$ $\cot^{2}\theta+1=\csc^{2}\theta$ because of the Pythagorean Identities $$=\frac{\csc^{2}\theta}{\cot\theta}$$ Rewrite using Reciprocal Identities: $$=\frac{1}{\sin^{2}\theta}\div\frac{\cos\theta}{\sin\theta}$$ $$=\frac{1}{\sin^{2}\theta}\times\frac{\sin\theta}{\cos\theta}$$ $$=\frac{1}{\sin\theta\cos\theta}=\csc\theta\sec\theta$$
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