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Chapter 7: Problem 8

Simplify the expressions. $$ \cos ^{2}(6 x)-\sin ^{2}(6 x) $$

Short Answer

Expert verified
The expression simplifies to \( \cos(12x) \).

Step by step solution

01

Identify the Trigonometric Identity

The expression \( \cos^2(6x) - \sin^2(6x) \) can be identified as an application of the double angle identity for cosine. Recall that the identity \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \) can be used in cases like this.
02

Apply the Double Angle Identity

Recognize that the expression \( \cos^2(6x) - \sin^2(6x) \) matches the form of \( \cos(2\theta) \). Here, \( \theta = 6x \). Thus, the expression simplifies using the identity: \( \cos(2\times 6x) = \cos(12x) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Double Angle Identity
The double angle identities are important tools in trigonometry. These identities allow you to express trigonometric functions with doubled angles in terms of single angles. For cosine, the double angle identity is \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \). This identity helps simplify expressions that appear complex at first glance.
  • Usefulness: By identifying parts of an expression that match this identity, you can simplify and reduce the complexity of the problem.
  • Application: The double angle formula applies when a trigonometric expression can be rewritten as products or squares of sine and cosine functions.
Take the expression \( \cos^2(6x) - \sin^2(6x) \). Recognize it as \( \cos(2 \cdot 6x) \), making the simplification straightforward with this identity.
Simplification
Simplification in mathematics involves reducing an expression to its most basic form. This process aims to make the expression easier to handle and understand. In trigonometry, simplification often uses identities to transform the expression.
  • Identify: Look for recognizable identities or patterns within the trigonometric expression.
  • Transform: Apply these identities to rewrite the expression in a simpler form.
When simplifying \( \cos^2(6x) - \sin^2(6x) \), we identify it as a double angle identity. By transforming it to \( \cos(12x) \), you effectively simplify the expression by using known trigonometric principles.
Cosine Formula
The cosine formula, including the double angle identity, is significant in simplifying expressions involving cosine. The identity \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \) is derived from the Pythagorean identity and the properties of cosine.
  • Concept: It shows how cosine can be expressed in terms of squared trigonometric components of an angle.
  • Application: Useful in scenarios where other methods of simplification are challenging or complex.
The expression \( \cos^2(6x) - \sin^2(6x) \) directly applies the cosine formula, revealing its simplicity as \( \cos(12x) \). This showcases the power of trigonometric identities in problem-solving.

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Most popular questions from this chapter

A population of deer oscillates 15 above and below average during the year, hitting the lowest value in January. The average population starts at 800 deer and increases by 110 each year. Find an equation for the population, \(P\), in terms of the months since January, \(t\).

Solve each equation for all solutions. \(\cos (2 x) \cos (x)+\sin (2 x) \sin (x)=1\)

Find all solutions on the interval \([0,2 \pi)\) \(2 \sin (x) \cos (x)-\sin (x)+2 \cos (x)-1=0\)

Solve for the first two positive solutions \(-5 \sin (x)+3 \cos (x)=1\)

Use the double angle, half angle, or power reduction formula to rewrite without exponents. $$ \cos ^{2} x \sin ^{4} x $$
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