Problem 70 - 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. \( \sec^3 x - \sec^2 x - \sec x + 1 \)

Step-by-Step Solution

Answer

The factorized and simplified form of the given expression is \( ((1/cos(x))-1)(tan^2(x)) \)

Step 1: Initial Analysis

Look at the expression \( \sec^3 x - \sec^2 x - \sec x + 1 \). It resembles a polynomial equation in terms of sec(x). Treat sec(x) as a variable, say 'u'. Therefore, the expression becomes \( u^3 - u^2 - u + 1 \).

Step 2: Factorize the Expression

This is a standard cubic equation of the form \( u^3 - u^2 - u + 1 \) which can be written as \( (u-1)(u^2-1) \). Substituting back 'u' as sec(x), the expression becomes \( (sec(x)-1)(sec^2(x)-1) \).

Step 3: Apply Pythagorean Identity

The Pythagorean Identity is \( sin^2(x) + cos^2(x) = 1 \). From which we have \( sec^2(x) = 1 + tan^2(x) \). Subsitute this into the equation to simplify further. The expression now becomes \( (sec(x)-1)(tan^2(x)) \)

Step 4: Apply Reciprocal Identity

The Reciprocal Identity is \( sec(x) = 1/cos(x) \). Subsitute this into the equation again to get the final simplified form. We finally get \( ((1/cos(x))-1)(tan^2(x)) \)

Key Concepts

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into simpler, multiply-able parts or factors.
Similar to the way that numbers can be factored into prime numbers, polynomials can be expressed as products of other polynomials.
This can make them easier to work with, especially when solving equations.

• Consider the polynomial expression \( \sec^3 x - \sec^2 x - \sec x + 1 \).
• If we treat \( \sec(x) \) as a variable, say \( u \), it transforms the expression into \( u^3 - u^2 - u + 1 \).
• This change of variable allows us to apply typical polynomial factorization techniques.

Breaking it down step by step, we first identify any common factors and use polynomial identities, like the difference of squares or cubes, to simplify expressions.
In this case, it factors nicely into \( (u-1)(u^2-1) \), which can be further broken down if needed.
When we substitute back \( sec(x) \), our expression then becomes \( (\sec(x)-1)(\sec^2(x)-1) \), an easier form to manipulate further.

Pythagorean Identities

Pythagorean identities are key relationships in trigonometry derived from the Pythagorean theorem.
These identities relate the square of the sine, cosine, and tangent functions.
One of the most familiar is \( \sin^2(x) + \cos^2(x) = 1 \).

• This identity can be rearranged to express other trigonometric functions in terms of one another, which is helpful in transformation and simplification processes.
• For our exercise, knowing that \( \sec^2(x) = 1 + \tan^2(x) \) allows us to replace the \( \sec^2(x) \) term.
• Using this identity, we rewrite \( (\sec^2(x)-1) \) as \( \tan^2(x) \), simplifying the expression further.

By converting \( \sec^2(x) \) in terms of \( \tan^2(x) \), we leverage the power of these identities to transform and condense expressions for easier manipulation.

Reciprocal Identities

Reciprocal identities relate trigonometric functions to their reciprocals. These identities are fundamental for simplifying expressions and solving equations in trigonometry.

• Specifically, the reciprocal identity for secant is \( \sec(x) = \frac{1}{\cos(x)} \).
• This identity is crucial when simplifying expressions to their most reduced form.

In our problem, replacing \( \sec(x) \) with \( \frac{1}{\cos(x)} \) provides the final step in simplification.
Given the expression \( ((1/\cos(x))-1)(\tan^2(x)) \), each term now reflects a basic trigonometric function, facilitating further understanding and application towards solving or simplifying the given equation.
The reciprocals help encapsulate the complexity and present it in a form easily digestible.