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Chapter 5: Problem 46

Perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer. $$\frac{1}{\sec x+1}-\frac{1}{\sec x-1}$$

Short Answer

Expert verified
The simplified form of the given equation is \( \frac{2 \sin^2 x - 2}{\sin^2 x} \)

Step by step solution

01

Rewrite using Reciprocal Identity

Rewrite \( \sec x \) as \( \frac{1}{\cos x} \). The equation now looks like: \( \frac{1}{\frac{1}{\cos x} + 1} - \frac{1}{\frac{1}{\cos x} - 1} \)
02

Simplify

Simplify each fraction in the equation by multiplying the numerator and the denominator by \( \cos x \). This results in: \( \frac{\cos x}{1+\cos x} - \frac{\cos x}{1-\cos x} \)
03

Find Common Denominator

Add the two fractions together by finding a common denominator, which is \( (1 + \cos x)(1 - \cos x) = 1 - \cos^2 x \). This simplifies to:\( \frac{\cos x(1 - \cos x) - \cos x(1 + \cos x)}{1 - \cos^2 x} \)
04

Further Simplify

Continuing to simplify the above expression results in: \( \frac{-2 \cos^2 x}{1 - \cos^2 x} \)
05

Apply Pythagorean Identity

Using the fundamental Pythagorean identity \( \cos^2 x + \sin^2 x = 1 \), we can replace \( \cos^2 x \) with \( 1 - \sin^2 x \) in the numerator and similarly for the denominator. This leads to:\( \frac{-2(1-\sin^2 x)}{\sin^2 x} \)
06

Final Simplification

Simplify this to get the final answer - the simplified form of the equation:\( \frac{2 \sin^2 x - 2}{\sin^2 x} \)

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