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Chapter 5: Problem 63

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
The simplified form of the given expression is 1.

Step by step solution

01

Substitute the Pythagorean identity for secant

Replace \( \sec^2(x) \) in the given expression with \( 1 + \tan^2(x) \). This will give us \( \frac{1+\tan^2(x) -1}{\sec x - 1} \).
02

Simplify the fraction

In the numerator, the 1's will cancel out, leaving only \( \tan^2(x) \). So, the simplified expression is \( \frac{\tan^2(x)}{\sec x - 1} \).
03

Factor the denominator

Factor the denominator \( \sec x - 1 \) by applying the Pythagorean identity for secant again to get \( \tan^2(x) \).
04

Simplify the expression

Now both the numerator and denominator are \( \tan^2(x) \), hence they cancel each other out, resulting in 1.

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Most popular questions from this chapter

Find all solutions of the equation in the interval \([0,2 \pi)\). $$2 \sec ^{2} x+\tan ^{2} x-3=0$$

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

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