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Chapter 5: Problem 24

Factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer. $$\frac{\cos x-2}{\cos ^{2} x-4}$$

Short Answer

Expert verified
The simplified version of the given trigonometric expression is \( \frac{1}{\sqrt{1-\sin^{2} x} + 2} \)

Step by step solution

01

Factor the denominator

The given expression is \( \frac{\cos x-2}{\cos^{2} x-4} \). This is a quadratic expression in the denominator. The denominator \( \cos^{2} x-4 \) can be factored using the difference of squares formula, giving \( (\cos x - 2)(\cos x + 2) \). The expression becomes \( \frac{\cos x-2}{(\cos x - 2)(\cos x + 2)} \)
02

Simplify the expression

We can now cancel out \( \cos x - 2 \) in the numerator and the denominator to get \( \frac{1}{\cos x + 2} \)
03

Further Simplify Using Pythagorean Identity

Replacing \( \cos x \) with \( \sqrt{1-\sin^{2} x} \), the expression becomes \( \frac{1}{\sqrt{1-\sin^{2} x} + 2} \)

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