Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

6. Trigonometric Identities and More Equations

Introduction to Trigonometric Identities

Trigonometry

6. Trigonometric Identities and More Equations

Introduction to Trigonometric Identities

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Problem 5.48

Textbook Question

Verify that each equation is an identity.
(tan² α + 1)/ sec α = sec α

Verified step by step guidance

Start by recalling the Pythagorean identity: \( \tan^2 \alpha + 1 = \sec^2 \alpha \).

Substitute \( \sec^2 \alpha \) for \( \tan^2 \alpha + 1 \) in the left-hand side of the equation, giving \( \frac{\sec^2 \alpha}{\sec \alpha} \).

Simplify the expression \( \frac{\sec^2 \alpha}{\sec \alpha} \) by canceling one \( \sec \alpha \) from the numerator and the denominator, resulting in \( \sec \alpha \).

Observe that the simplified left-hand side \( \sec \alpha \) is equal to the right-hand side \( \sec \alpha \).

Conclude that the original equation \( \frac{\tan^2 \alpha + 1}{\sec \alpha} = \sec \alpha \) is indeed an identity.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. Common identities include the Pythagorean identities, reciprocal identities, and quotient identities. Understanding these identities is crucial for verifying equations and simplifying expressions in trigonometry.

Tangent and Secant Functions

The tangent function, tan(α), is defined as the ratio of the opposite side to the adjacent side in a right triangle, or as sin(α)/cos(α). The secant function, sec(α), is the reciprocal of the cosine function, defined as 1/cos(α). Recognizing the relationships between these functions is essential for manipulating and verifying trigonometric equations.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. This includes factoring, expanding, and combining like terms. In the context of trigonometric identities, effective algebraic manipulation allows one to transform one side of an equation to match the other, thereby verifying the identity.