Problem 64 - Precalculus with Limits

Question

In Exercises 59 - 70, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer. \( \dfrac{\cos^2 x - 4}{\cos x - 2} \)

Step-by-Step Solution

Answer

The simplified form of \( \dfrac{\cos^2 x - 4}{\cos x - 2} \) is \( \cos x + 2 \) given \( \cos x \neq 2 \).

Step 1: Factorize the expression in the numerator

Recognizing that the numerator \( \cos^2 x - 4 \) is a difference of two squares, it can be factored as follows: \( (\cos x - 2)(\cos x + 2) \). So the given expression becomes \( \frac{(\cos x - 2)(\cos x + 2)}{\cos x - 2} \)

Step 2: Simplify the expression

Since \( \cos x - 2 \) is present both in numerator and denominator, they can be cancelled out. This simplifies the expression to: \( \cos x + 2 \)

Step 3: Consider the domain restrictions

The simplified form \( \cos x + 2 \) is only valid if \( \cos x \neq 2 \). Therefore, all the values of \( x \) for which \( \cos x \neq 2 \) should be taken into account when dealing with the simplified form.

Key Concepts

Factoring Expressions

Factoring expressions is a key skill in algebra and trigonometry. It involves breaking down an expression into simpler parts or factors that can be multiplied together to yield the original expression. In the given exercise, the expression in the numerator, \( \cos^2 x - 4 \), is factored using the method known as the "Difference of Squares." To factor, you should recognize common patterns or identities. Here, the expression \( \cos^2 x - 4 \) matches the format \( a^2 - b^2 \), which is a difference of squares. This pattern can be rewritten as two multiplied binomials: \((a - b)(a + b)\). Breaking down the differences of squares:

• \( a^2 = \cos^2 x \)
• \( b^2 = 4 \) or \( 2^2 \)

Applying the difference of squares, you get \((\cos x - 2)(\cos x + 2)\). This has simplified the original numerator into factors that are much easier to work with. Properly factoring expressions like this one simplifies calculations and makes solving equations more manageable.

Difference of Squares

The difference of squares is a common technique used to factor certain types of algebraic expressions. It's particularly useful when you have terms in the form \( a^2 - b^2 \). This can be simplified into \((a - b)(a + b)\). Understanding and recognizing this pattern allows you to simplify and solve complex expressions with ease. In the exercise provided, the expression \( \cos^2 x - 4 \) was identified as a difference of squares:

• First, identify \( a^2 \), which is \( \cos^2 x \).
• Next, identify \( b^2 \), which is \( 4 \), equivalent to \( 2^2 \).

This results in the factors \((\cos x - 2)(\cos x + 2)\), simplifying both the process of manipulation and the resulting expression. Practicing this technique enhances your ability to spot and resolve such expressions in various mathematical contexts. Use it whenever possible to condense and clarify algebraic expressions.

Domain Restrictions

Domain restrictions are an important concept to consider when simplifying expressions and solving equations. A domain restriction typically indicates the set of input values \( x \) for which the expression or function is defined. For the simplified expression \( \cos x + 2 \), derived from the original problem, the crucial aspect to keep in mind is the elimination step during factor cancellation. Even if factors reduce the fraction, constraints still apply—detailed by the restrictions on the denominator. When you cancel \( \cos x - 2 \) from both numerator and denominator, this sets a condition: you cannot let \( \cos x \) equal 2, as it would originally cause the denominator to be zero.The requirement that \( \cos x eq 2 \) arises since cosine ranges between -1 and 1 for real numbers; thus, \( \cos x \) can never reach 2, but denominators typically deal with theoretical behavior. Always consider such restrictions when you simplify, ensuring your final expression is valid and complete.