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 5.40

Textbook Question

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
csc² t - 1

Verified step by step guidance

Recognize that the expression \( \csc^2 t - 1 \) can be related to a fundamental trigonometric identity.

Recall the Pythagorean identity: \( \csc^2 t = 1 + \cot^2 t \).

Substitute \( \csc^2 t \) in the expression with \( 1 + \cot^2 t \) to get \( (1 + \cot^2 t) - 1 \).

Simplify the expression by subtracting 1: \( \cot^2 t \).

Conclude that the simplified expression is \( \cot^2 t \), which is a power of a trigonometric function.

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Key Concepts

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

Cosecant Function

The cosecant function, denoted as csc(t), is the reciprocal of the sine function. It is defined as csc(t) = 1/sin(t). Understanding this function is crucial for simplifying expressions involving csc²(t), as it relates directly to the sine function and its properties.

Pythagorean Identity

The Pythagorean identities are fundamental relationships in trigonometry that relate the squares of the sine and cosine functions. One key identity is sin²(t) + cos²(t) = 1. This identity can be rearranged to express csc²(t) in terms of sin²(t), which is essential for simplifying expressions like csc²(t) - 1.

Fundamental Trigonometric Identities

Fundamental trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where the functions are defined. These include reciprocal identities, Pythagorean identities, and co-function identities. Utilizing these identities allows for the simplification of complex trigonometric expressions into more manageable forms.

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Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.csc² t - 1

Key Concepts

Cosecant Function

Pythagorean Identity

Fundamental Trigonometric Identities

Watch next

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
csc² t - 1