Algebra and Trigonometry, 5th Edition

7.1 Identities: Pythagorean and Sum and Difference

• State the Pythagorean identities.

• Simplify and manipulate expressions containing trigonometric expressions.

• Use the sum and difference identities to find function values.

An identity is an equation that is true for all possible replacements of the variables. The following is a list of the identities studied in Chapter 6.

Basic Identities

$$\begin{array}{ccc}\hfill \text{sin} x={\displaystyle \frac{1}{\text{csc} x}},& \text{csc} x={\displaystyle \frac{1}{\text{sin} x}},\hfill & \text{sin}\left(-x\right)=-\text{sin}x,\hfill \\ \text{cos} \left(-x\right)=\text{cos} x,\hfill \\ \hfill \text{cos} x={\displaystyle \frac{1}{\text{sec} x}},& \text{sec} x={\displaystyle \frac{1}{\text{cos} x}},\hfill & \text{tan} \left(-x\right)=-\text{tan}x,\hfill \\ \hfill \text{tan} x={\displaystyle \frac{1}{\text{cot} x}},& \text{cot} x={\displaystyle \frac{1}{\text{tan} x}},\hfill & \text{tan} x={\displaystyle \frac{\text{sin} x}{\text{cos} x}},\hfill \\ \text{cot} x={\displaystyle \frac{\text{cos} x}{\text{sin} x}}\hfill \end{array}$$

In this section, we will develop some other important identities.

Pythagorean Identities

We now consider three other identities that are fundamental to a study of trigonometry. They are called ...