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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2:44 minutes

Problem 37

Textbook Question

In Exercises 35–38, use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1.sin² x cos² x

Verified step by step guidance

Identify the power-reducing formulas for \( \sin^2 x \) and \( \cos^2 x \): \( \sin^2 x = \frac{1 - \cos(2x)}{2} \) and \( \cos^2 x = \frac{1 + \cos(2x)}{2} \).

Substitute the power-reducing formulas into the expression \( \sin^2 x \cos^2 x \).

The expression becomes \( \left( \frac{1 - \cos(2x)}{2} \right) \left( \frac{1 + \cos(2x)}{2} \right) \).

Simplify the expression by multiplying the two fractions: \( \frac{(1 - \cos(2x))(1 + \cos(2x))}{4} \).

Use the difference of squares formula: \( (a - b)(a + b) = a^2 - b^2 \), to simplify further to \( \frac{1 - \cos^2(2x)}{4} \).

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Key Concepts

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

Power-Reducing Formulas

Power-reducing formulas are trigonometric identities that express powers of sine and cosine in terms of the first power of sine and cosine. These formulas are particularly useful for simplifying expressions involving sin²(x) and cos²(x). For example, sin²(x) can be rewritten as (1 - cos(2x))/2, and cos²(x) can be expressed as (1 + cos(2x))/2.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variables involved. They are fundamental in simplifying trigonometric expressions and solving equations. Key identities include the Pythagorean identities, reciprocal identities, and co-function identities, which provide relationships between different trigonometric functions.

Simplification of Trigonometric Expressions

Simplification of trigonometric expressions involves rewriting complex expressions into simpler forms, often using identities. This process is essential for solving trigonometric equations or integrating trigonometric functions. By applying power-reducing formulas, one can reduce the degree of the trigonometric functions, making calculations more manageable.