Problem 69 - 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. \( \csc^3 x - \csc^2 x - \csc x + 1 \)

Step-by-Step Solution

Answer

The simplest form of the given expression is \(((\cos^2 x)/\sin^2 x)*(\sec x)\)

Step 1: Factor the expression

Notice that the expression \( \csc^3 x - \csc^2 x - \csc x + 1 \) is in the form of a cubic equation. The first step to simplifying it is by factoring. It can be recognized as a difference of cubes and factored in this way: \( (\csc x -1)^2 * (\csc x + 1)\)

Step 2: Apply the reciprocal identity

The reciprocal identity for cosecant is \(\csc x = 1/\sin x\), so replace \(\csc x\) in the factored expression with \(1/\sin x\), yielding: \( (1/\sin x - 1)^2 * (1/\sin x + 1)\)

Step 3: Simplify the expression

Rearrange and simplify the above expression to get it to its simplest form. First, combine the fraction terms in each of the binomial expressions, then simplify the squared terms and multiply that with the third term to get the final simplified expression: \( ( (1 - \sin x)/\sin x)^2 * ((1 + sin x)/sin x) = ((\cos^2 x)/\sin^2 x)*(\sec x)\)

Key Concepts

Understanding Reciprocal Identity

In trigonometry, reciprocal identities help to express one trigonometric function in terms of another. The cosecant function, denoted as \( \csc x \), is the reciprocal of the sine function. This means:

• \( \csc x = \frac{1}{\sin x} \)

This identity is extremely useful in simplifying trigonometric expressions because it allows you to convert a function into another that may be easier to work with.
For example, if you encounter \( \csc x \) in an expression and need to factor or simplify further, using \( \frac{1}{\sin x} \) can sometimes make other approaches available. Reciprocals often simplify the math by converting division into simple multiplication or preparing terms for factoring.
Understanding this concept is crucial when working with trigonometric equations, as it often forms the basis for simplifying complex mathematical expressions.

Exploring the Difference of Cubes

A difference of cubes refers to an expression of the form \( a^3 - b^3 \). The special formula to factor this expression is:

• \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)

In this exercise, the expression \( \csc^3 x - \csc^2 x - \csc x + 1 \) can resemble forms that use this identity. Notice how factoring works to split the polynomial into these parts, making it easier to handle.
Here, the solution factored it as \( (\csc x -1)^2 (\csc x + 1) \). This process involves re-arranging and simplifying until the expression matches the recognizable pattern of a cubic difference or resembles it closely.
Becoming comfortable with this factoring method equips you with a powerful tool for solving polynomial equations and simplifying expressions.

A Closer Look at Cosecant

Cosecant is one of the six fundamental trigonometric functions, often symbolized by \( \csc x \). It is especially significant because it is the reciprocal of sine, often used in solving and simplifying trigonometric problems.

• \( \csc x = \frac{1}{\sin x} \)
• Provides the vertical stretch for the sine wave's reciprocal.

Cosecant appears frequently in the solutions for trigonometric identities and equations. Understanding cosecant aids in working through more complex expressions, as seen when factoring or manipulating trigonometric cubes like \( \csc^3 x \).
This function is not defined for the integers of \( n\pi \), where \( n \) is an integer, because at those points, sine equals zero, and division by zero is undefined.
Gaining a strong hold of the cosecant's behavior strengthens your grasp of trigonometric concepts and complements working with sine and other related functions.