Algebra and Trigonometry, 5th Edition

Chapter 7 Summary and Review

Study Guide

KEY TERMS AND CONCEPTS	EXAMPLES
SECTION 7.1: IDENTITIES: PYTHAGOREAN AND SUM AND DIFFERENCE
Identity An identity is an equation that is true for all possible replacements. Basic Identities $\text{sin}x={\displaystyle \frac{1}{\text{csc}x}},\text{tan}x={\displaystyle \frac{\text{sin}x}{\text{cos}x}}$, $\text{cos}x={\displaystyle \frac{1}{\text{sec}x}},\text{cot}x={\displaystyle \frac{\text{cos}x}{\text{sin}x}}$, $\text{tan}x={\displaystyle \frac{1}{\text{cot}x}}$, $\text{sin}\left(-x\right)=-\text{sin}x$, $\text{cos}\left(-x\right)=\text{cos}x$, $\text{tan}\left(-x\right)=-\text{tan}x$ Pythagorean Identities ${\text{sin}}^{2}x+{\text{cos}}^{2}x=1$, $1+{\text{cot}}^{2}x={\text{csc}}^{2}x$, $1+{\text{tan}}^{2}x={\text{sec}}^{2}x$	Simplify. a) ${\displaystyle \frac{{\text{tan}}^{2}x\text{sec}x}{{\text{sec}}^{2}x\text{tan}x}}$ b) ${\displaystyle \frac{{\text{cos}}^2\alpha -2\text{cos}\alpha -15}{\text{cos}\alpha +3}}$ c) ${\displaystyle \frac{\text{cos}\theta }{\text{sin}\left(-\theta \right)}}+{\displaystyle \frac{1}{\text{tan}\theta }}$ d) $\sqrt{{\displaystyle \frac{1+\text{sin}x}{1-\text{sin}x}}}$ a) ${\displaystyle \frac{{\text{tan}}^{2}x\text{sec}x}{{\text{sec}}^{2}x\text{tan}x}}={\displaystyle \frac{\text{tan}x}{\text{sec}x}}={\displaystyle \frac{\text{sin}x}{\text{cos}x}}\cdot {\displaystyle \frac{\text{cos}x}{1}}=\text{sin}x$ b) ${\displaystyle \frac{{\text{cos}}^2\alpha -2\text{cos}\alpha -15}{\text{cos}\alpha +3}}={\displaystyle \frac{\left(\text{cos}\alpha -5\right)\left(\text{cos}\alpha +3\right)}{\text{cos}\alpha +3}}=\text{cos}\alpha -5$ c) ${\displaystyle \frac{\text{cos}\theta }{\text{sin}\left(-\theta \right)}}+{\displaystyle \frac{1}{\text{tan}\theta }}=-{\displaystyle \frac{\text{cos}\theta }{\text{sin}\theta }}+{\displaystyle \frac{\text{cos}\theta }{\text{sin}\theta }}=0$ d) $\begin{array}{c}\sqrt{{\displaystyle \frac{1+\text{sin}x}{1-\text{sin ...}}}}\hfill \end{array}$