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Chapter 5: Problem 70

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 expression \(\sec^{3} x-\sec^{2} x-\sec x+1\) can be factored and simplified to \((\sec x - 1) \tan^{2} x \).

Step by step solution

01

Factor by Grouping

This is a four-term polynomial and can be therefore factored by grouping. Group the terms two at a time. Group the first two terms and the last two terms as follows: \( \sec^{3} x-\sec^{2} x \) and \(-\sec x+1\). Factoring out the greatest common factor in both pairs gives \( \sec^{2} x(\sec x - 1) \) and \( -1(\sec x - 1)\).
02

Factoring out the common binomial

Our grouping from Step 1 resulted in two terms. Notice a common binomial \( (\sec x - 1) \) in both terms. Factoring this out gives \( (\sec x - 1) (\sec^{2} x - 1) \).
03

Applying the Pythagorean Identity

The part of the factored expression, \( \sec^{2} x - 1 \), can be identified as a form of the Pythagorean identity \( \sec^{2} x = 1 + \tan^{2} x \) or \( \sec^{2} x - 1 = \tan^{2} x \). Replacing \( \sec^{2} x - 1 \) with \( \tan^{2} x \) gives a simplified form of the expression as \( (\sec x - 1) \tan^{2} x \).

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