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Chapter 5: Problem 23

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

Short Answer

Expert verified
Two possible simplified expressions for \(\frac{\sec^{2} x - 1}{\sec x - 1}\) are \(\frac{\sin^{2} x}{\cos x (1 - \cos x)}\) and \(\frac{\tan^{2} x}{\frac{1}{\cos x} -1}\)

Step by step solution

01

Apply Fundamental Identities

In order to simplify, we replace the \(\sec^{2} x\) with \(\tan^{2}x + 1\) using the identity \(\sec^{2} x = \tan^{2} x + 1\). The expression now becomes \(\frac{\tan^{2}x + 1 - 1}{\sec x - 1}\)
02

Simplify the Numerator

The numerator simplifies to \(\tan^{2}x\), therefore the expression becomes \(\frac{\tan^{2}x}{\sec x - 1}\)
03

Apply Fundamental Identities Again

Next, we again replace \(\sec x\) with \(\frac{1}{\cos x}\) and \(\tan x\) with \(\frac{\sin x}{\cos x}\). Therefore, the resulting expression is \(\frac{(\frac{\sin^{2} x}{\cos^{2} x})}{(\frac{1}{\cos x} - 1)}\)
04

Simplifying Expression

We multiply every term in the fraction by \(\cos^{2} x\) to get \(\frac{\sin^{2} x}{\cos x - \cos^{2} x}\)
05

Factor the Denominator

Factoring the denominator by \(\cos x\), we get \(\frac{\sin^{2} x}{\cos x (1 - \cos x)}\) which is a simplified form of the original expression. It can be further simplified into different forms depending on the context needed.

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