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

Sum and Difference Identities

Trigonometry

6. Trigonometric Identities and More Equations

Sum and Difference Identities

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Problem 5.73

Textbook Question

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)
cos 2x = (cot² x - 1)/(cot² x + 1)

Verified step by step guidance

Start by using the hint: express \( \cos 2x \) as \( \cos(x + x) \). Use the cosine addition formula: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \).

Substitute \( a = x \) and \( b = x \) into the formula: \( \cos 2x = \cos x \cos x - \sin x \sin x = \cos^2 x - \sin^2 x \).

Recall the Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \). From this, express \( \sin^2 x \) as \( 1 - \cos^2 x \).

Substitute \( \sin^2 x = 1 - \cos^2 x \) into the expression for \( \cos 2x \): \( \cos 2x = \cos^2 x - (1 - \cos^2 x) = 2\cos^2 x - 1 \).

Now, express \( \cot^2 x \) in terms of \( \cos x \) and \( \sin x \): \( \cot x = \frac{\cos x}{\sin x} \), so \( \cot^2 x = \frac{\cos^2 x}{\sin^2 x} \). Substitute this into the right side of the given identity and simplify to verify the 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. They are fundamental in simplifying expressions and solving equations in trigonometry. Common identities include the Pythagorean identities, angle sum and difference identities, and double angle formulas, which are essential for verifying equations like the one presented.

Double Angle Formulas

Double angle formulas express trigonometric functions of double angles in terms of single angles. For example, the cosine double angle formula states that cos(2x) can be expressed as cos²(x) - sin²(x) or 2cos²(x) - 1. Understanding these formulas is crucial for transforming and verifying trigonometric equations, such as the one given in the question.

Cotangent Function

The cotangent function, denoted as cot(x), is the reciprocal of the tangent function, defined as cot(x) = cos(x)/sin(x). It plays a significant role in trigonometric identities and can be expressed in terms of sine and cosine. In the context of the given equation, recognizing how cotangent relates to other trigonometric functions is essential for verifying the identity.