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Chapter 5: Problem 66

Factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer. $$1-2 \cos ^{2} x+\cos ^{4} x$$

Short Answer

Expert verified
The simplified and factored form of the given expression is \( (\sin^2x - 1)^2\).

Step by step solution

01

Identify structure of the expression

The expression is of form \(a-bx+x^2\) which is a standard quadratic equation that can be factored or simplified.
02

Rewrite expression using fundamental identities

Replace \(\cos^4x\) with \((\cos^2x)^2\) then apply the Pythagorean identity \(\cos^2x = 1 - \sin^2x\). Therefore, our expression becomes \(1 - 2(1-\sin^2x) + (1 - \sin^2x)^2\).
03

Simplify the expression

Expand and combine like terms to get \(\sin^4x - 2\sin^2x + 1\).
04

Factor the expression

The expression is a perfect square trinomial \( (\sin^2x - 1)^2\).

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Most popular questions from this chapter

Write the expression as the sine, cosine, or tangent of an angle. $$\sin 60^{\circ} \cos 15^{\circ}+\cos 60^{\circ} \sin 15^{\circ}$$

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

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the interval \([0,2 \pi)\). $$x \cos x-1=0$$

Solve the multiple-angle equation. $$\sec 4 x=2$$

Solve the multiple-angle equation. $$\cos 2 x=\frac{1}{2}$$
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