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Chapter 5: Problem 28

Factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer. $$\sec ^{3} x-\sec ^{2} x-\sec x+1$$

Short Answer

Expert verified
The simplified form of the expression \( \sec ^{3} x-\sec ^{2} x-\sec x+1 \) is \( (\tan^2 x)(\sec x -1)\).

Step by step solution

01

Recognize the quadratic form

Recognize the expression as a quadratic in terms of \( \sec x \). The given expression, \( \sec^3 x - \sec^2 x - \sec x + 1 \), can be written in terms of \( \sec x \) i.e. \( a(\sec x)^2 + b(\sec x) + c \) where a = 1, b = -1, c = -1.
02

Factor the quadratic expression

The quadratic expression can be factored just like a quadratic equation. By grouping, it can be expressed as \( (\sec^2 x -1)(\sec x -1)\).
03

Apply fundamental identities

Now, we will use the identity \( \sec^2 x - 1 = \tan^2 x\) to replace \( \sec^2 x - 1\) in the expression. Thus it simplifies the expression to \( (\tan^2 x)(\sec x -1)\).
04

Simplify the expression

This is the final simplified form of the original expression. There could be multiple forms due to the multiple representations for sec and tan based on sine, cosine, and other trigonometric identities.

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