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Chapter 5: Problem 68

Verify the identity. $$\cos ^{4} x-\sin ^{4} x=\cos 2 x$$

Short Answer

Expert verified
Using trigonometric identities and transformations, it can be shown that \(\cos^4x - \sin^4x\) simplifies to \(\cos2x\), thus verifying the given identity.

Step by step solution

01

Express cos^4 and sin^4 in terms of cos^2 and sin^2

The first step is to realize that \(\cos^4x\) can be written as \((\cos^2x)^2\) and \(\sin^4x\) can be written as \((\sin^2x)^2\). This gives us \((\cos^2x)^2 - (\sin^2x)^2\).
02

Apply the difference of squares formula

Notice that the expression \((\cos^2x)^2 - (\sin^2x)^2\) can be factored using the difference of squares formula \((a^2 - b^2) = (a+b)(a-b)\). This transform our expression into \((\cos^2x + \sin^2x)(\cos^2x - \sin^2x)\). Since \(\cos^2x + \sin^2x = 1\), this simplifies to \(\cos^2x - \sin^2x\)
03

Recognize and Apply the double-angle formula

The expression \(\cos^2x - \sin^2x\) should be recognized as the double-angle formula for cosine, specifically \(\cos2x\). Thus, \(\cos^2x - \sin^2x = \cos2x\) and the identity is verified.

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Key Concepts

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

Double Angle Formulas
The double angle formulas are a set of trigonometric identities that help simplify expressions involving trigonometric functions of angle sums and multiples. Double angle formulas can be particularly handy when working with cosine, sine, or tangent of twice an angle.
The double angle formula for cosine, which often comes up in these exercises, is given by:
  • \( \cos 2x = \cos^2 x - \sin^2 x \)
  • It can also be written as \( \cos 2x = 2\cos^2 x - 1 \) or \( \cos 2x = 1 - 2\sin^2 x \)
In our context, recognizing the formula \( \cos^2 x - \sin^2 x = \cos 2x \) helped verify the trigonometric identity. Double angle formulas are very useful in transforming complex trigonometric expressions into simpler forms.
Difference of Squares
The difference of squares is a key algebraic identity. It simplifies expressions of the form \( a^2 - b^2 \) into a product: \( (a + b)(a - b) \). This identity is useful because it turns a subtraction of squares into something that can easily be factored.
  • For example, \( x^2 - y^2 = (x + y)(x - y) \)
In the step-by-step solution of the identity \( \cos^4 x - \sin^4 x \), the difference of squares was applied to \( (\cos^2 x)^2 - (\sin^2 x)^2 \). This allowed the expression to transform into \( (\cos^2x + \sin^2x)(\cos^2x - \sin^2x) \). This step is crucial for simplifying and eventually solving the identity, making concepts like double angle formulas more approachable.
Cosine Function
The cosine function is one of the primary trigonometric functions, often denoted as \( \cos \theta \). It's fundamental in trigonometry and is used to describe the relation in a right triangle between the angle and the ratio of the adjacent side over the hypotenuse.
  • Cosine values vary from -1 to 1.
  • The function is periodic, with a period of \( 2\pi \).
  • Cosine is even: \( \cos(-x) = \cos(x) \).
In trigonometric identities, cosine often appears in various transformations like the double angle formula. Recognizing \( \cos^2 x - \sin^2 x \) as equal to \( \cos 2x \) shows the flexibility and importance of the cosine function in simplifying expressions and verifying identities. Understanding how to manipulate and apply cosine rules makes tackling trigonometric problems much easier.

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Most popular questions from this chapter

Use inverse functions where needed to find all solutions of the equation in the interval \(\mathbf{0}, \mathbf{2} \boldsymbol{\pi}\) ). $$\sec ^{2} x-4 \sec x=0$$

(a) use a graphing utility to graph the function and approximate the maximum and minimum points on the graph in the interval \(\mathbf{0}, \mathbf{2} \boldsymbol{\pi}\) ), and (b) solve the trigonometric equation and demonstrate that its solutions are the \(x\) -coordinates of the maximum and minimum points of \(f .\) (Calculus is required to find the trigonometric equation.) $$\begin{array}{cc}\text{Function} && \text {Trigonometric Equation} \\\ f(x)=\sin x \cos x && -\sin ^{2} x+\cos ^{2} x=0 \end{array}$$

Use inverse functions where needed to find all solutions of the equation in the interval \(\mathbf{0}, \mathbf{2} \boldsymbol{\pi}\) ). $$\csc ^{2} x+3 \csc x-4=0$$

The equation of a standing wave is obtained by adding the displacements of two waves traveling in opposite directions (see figure). Assume that each of the waves has amplitude \(A,\) period \(T,\) and wavelength \(\lambda .\) If the models for these waves are $$y_{1}=A \cos 2 \pi\left(\frac{t}{T}-\frac{x}{\lambda}\right) \text { and } y_{2}=A \cos 2 \pi\left(\frac{t}{T}+\frac{x}{\lambda}\right)$$ .Then show that $$y_{1}+y_{2}=2 A \cos \frac{2 \pi t}{T} \cos \frac{2 \pi x}{\lambda}$$.

Find all solutions of the equation in the interval \(0,2 \pi) .\) Use a graphing utility to graph the equation and verify the solutions. $$\cos 2 x-\cos 6 x=0$$
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