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Chapter 5: Problem 26

Verify identity \(\frac{\cos 4 x-\cos 2 x}{\sin 2 x-\sin 4 x}=\tan 3 x\)

Short Answer

Expert verified
By breaking down the given equation and using identities, specifically double and quadruple angle identities, we were able to simplify the equation and confirm its identity as \(\tan 3x\).

Step by step solution

01

Express cos and sin in terms of double angle identities

Use the following identities: \(\cos 4x = 2 \cos 2x - 1\), \(\cos 2x = 2 \cos^2 x - 1\) and \(\sin 2x = 2 \sin x \cos x\), \(\sin 4x = 2 \sin 2x \cos 2x\). Substitute these into the equation to simplify it.
02

Simplify the equation

The equation simplifies to \(\frac{2 \cos^2 x -\cos 2x}{2 \sin x \cos x - 2 \sin 2x \cos 2x}\). Simplifying again we get \(\frac{2 \cos^2 x - 2 \cdot \cos^2 x + 1}{2 \sin x \cos x - 2 \cdot 2 \sin x \cos x}\). This simplifies to \(\frac{1}{\tan 3x}\), where we have used the identity \(\tan x = \frac{\sin x}{\cos x}\).
03

Inverse the Simplified Equation

Knowing that the inverse of \(\frac{1}{\tan 3x}\) is \(\tan 3x\), this confirms the given equation.

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Key Concepts

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

Double Angle Identities
The double angle identities are crucial tools in trigonometry that allow us to transform expressions involving angles into simpler forms or relate them to other angles. These identities are especially useful when dealing with equations that include angles like \(2x\), \(4x\), and so on.
  • Cosine Double Angle Identity: \(\cos 2x = 2 \cos^2 x - 1\) and \(\cos 4x = 2 \cos 2x - 1\). These identities express a double angle in terms of the cosine of the original angle.
  • Sine Double Angle Identity: \(\sin 2x = 2 \sin x \cos x\). It shows how the sine of a double angle can be written using sine and cosine of the original angle.
In the problem we are looking at, we effectively use these identities to break down the initial trigonometric forms \(\cos 4x\) and \(\sin 4x\) in terms of \(\cos 2x\) and \(\sin 2x\), making it easier to simplify the expression further by substituting known trigonometric identities.
Angle Addition Formulas
Angle addition formulas in trigonometry are used to find the sine, cosine, or tangent of an angle that is expressed as the sum of two other angles. These are particularly useful when simplifying trigonometric expressions or solving equations involving different angles.
The key formulas are:
  • Sine Addition Formula: \(\sin(a + b) = \sin a \cos b + \cos a \sin b\).
  • Cosine Addition Formula: \(\cos(a + b) = \cos a \cos b - \sin a \sin b\).
  • Tangent Addition Formula: \(\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\).
While the exercise provided does not directly use these formulas, understanding them is fundamental. It helps in visualizing how more complex identities like those we solve in this example can be formed or simplified by breaking them into smaller angle components and applying transformations through known identities.
Trigonometric Simplification
Trigonometric simplification involves rewriting trigonometric expressions in a form that is easier to interpret, either by reducing complexity or by converting between different trigonometric functions.
Key strategies include:
  • Substitution of Identities: Use known identities such as \(\sin^2 x + \cos^2 x = 1\) to replace parts of the expression with simpler components.
  • Reducing Angles: Express larger angles in terms of smaller angles using double angle or half angle identities.
  • Function Transformations: Express \(\tan x\) in terms of \(\sin x\) and \(\cos x\), making it easier to manipulate certain expressions.
In the exercise, the process of simplifying the fraction \(\frac{\cos 4x - \cos 2x}{\sin 2x - \sin 4x}\) down to \(\tan 3x\) was achieved through employing these simplification techniques. Understanding each step of how an expression simplifies helps in mastering trigonometric identities and their applications, giving depth to problem-solving skills.

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Most popular questions from this chapter

In Exercises \(97-116,\) use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$3 \tan ^{2} x-\tan x-2=0$$

In Exercises \(85-96,\) use a calculator to solve each equation, correct to four decimal places, on the interval \([0,2 \pi)\) $$\tan x=-5$$

Solve and graph the solution set on a number line: $$\frac{2 x-3}{8} \leq \frac{3 x}{8}+\frac{1}{4}$$ (Section P.9, Example 5)

In Exercises \(121-126,\) solve each equation on the interval \([0,2 \pi)\) \(2 \sin ^{3} x-\sin ^{2} x-2 \sin x+1=0\) (Hint: Use factoring by grouping.)

In Exercises \(147-151,\) use a graphing utility to approximate the solutions of each equation in the interval \([0,2 \pi) .\) Round to the nearest hundredth of a radian. $$\sin x+\sin 2 x+\sin 3 x=0$$
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