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Chapter 5: Problem 45

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

Short Answer

Expert verified
The expression simplifies to \(2 \csc^2 x\).

Step by step solution

01

Add the two fractions

To add the fractions, find the common denominator. Here, the denominators are \(1+\cos x\) and \(1-\cos x\), hence, the common denominator is \((1+\cos x)(1-\cos x)\) which simplifies to \(1-\cos^2 x\) using the formula \(x^2 - y^2 = (x+y)(x-y)\). The numerator will be the sum of the products of the cross multiplication. Thus, the first step produces the fraction: \(\frac{1*(1-\cos x) + 1*(1+\cos x)}{1-\cos^2 x}\)
02

Simplify the fraction

Simplify the fraction by performing the operations in the numerator and the denominator. Adding the cross-multiplication products in the numerator, we get: \(1-\cos x + 1+\cos x\), which simplifies to \(2\). In the denominator, \(1-\cos^2 x\) is a fundamental identity that simplifies to \(\sin^2 x\), according to the Pythagorean identity. Thus, we get \(\frac{2}{\sin^2 x}\) as the simplified fraction.
03

Apply the reciprocal identity

In this step, apply the reciprocal identity for sine, which is that \(1/ \sin x = \csc x\). Thus, \(\frac{2}{\sin^2 x}\) simplifies to \(2 \csc^2 x\), and that's the final simplest form of the given trigonometric expression.

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