Algebra 2 Common Core

Published by Prentice Hall
ISBN 10: 0133186024
ISBN 13: 978-0-13318-602-4

Chapter 14 - Trigonometric Identities and Equations - 14-1 Trigonometric Identities - Practice and Problem-Solving Exercises: 29

Answer

$$\sec\theta$$

Work Step by Step

Simplify $$\cos\theta+\sin\theta\tan\theta$$ by substituting the Tangent Identity: $$\tan\theta=\frac{\sin\theta}{\cos\theta}$$ to obtain $$\cos\theta+\sin\theta\tan\theta=\cos\theta+\sin\theta\bigg(\frac{\sin\theta}{\cos\theta}\bigg)=\cos\theta+\frac{\sin^{2}\theta}{\cos\theta}$$ Now multiply both the numerator and the denominator of the first term by $\cos\theta$ $$=\cos\theta\bigg(\frac{\cos\theta}{\cos\theta}\bigg)+\frac{\sin^{2}\theta}{\cos\theta}$$ $$=\frac{\cos^{2}\theta+\sin^{2}\theta}{\cos\theta}$$ Next, use the Pythagorean Identity: $$\sin^{2}\theta+\cos^{2}\theta=1$$ and finally the Reciprocal Identity: $$\sec\theta=\frac{1}{\cos\theta}$$ to obtain $$\cos\theta+\sin\theta\tan\theta=\frac{1}{\cos\theta}=\sec\theta$$
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