Problem 57 - Precalculus with Limits

Question

In Exercises 53-60, (a) use a graphing utility to graph each side of the equation to determine whether the equation is an identity, (b) use the table feature of a graphing utility to determine whether the equation is an identity, and (c) confirm the results of parts (a) and (b) algebraically. \( \csc^4 x - 2 \csc^2 x + 1 = \cot^4 x \)

Step-by-Step Solution

Answer

The equation \(\csc^4 x - 2 \csc^2 x + 1 = \cot^4 x\) is an identity, as shown by both the graphing and algebraic method.

Step 1: Graphing the functions

Use a graphing utility to sketch the graphs for \(y = \csc^4 x - 2 \csc^2 x + 1\) and \(y = \cot^4 x\). Carefully analyze and compare both of these graphs.

Step 2: Using the Table Feature

Use the table feature of the graphing utility. List out the \(x\)-values and corresponding \(y\)-values for both functions \(y = \csc^4 x - 2 \csc^2 x + 1\) and \(y = \cot^4 x\) at the same \(x\)-values. Compare the outcome of the tables for both the functions.

Step 3: Algebraic Confirmation

To confirm the results, use the trigonometric identity \(\csc^2 x = 1 + \cot^2 x\) to express the left side of the equation in terms of cotangent. \(\csc^4 x - 2 \csc^2 x + 1 = (\cot^2x + 1)^2 - 2 (\cot^2x+1) +1\). Simplify the equation to receive \(\cot^4x + 1 -2\cot^2x +\cot^4x -2\cot^2x +1 -2 = \cot^4 x\). Thus, it is confirmed that the expression is indeed an identity.

Key Concepts

Cosecant Function

The cosecant function, often abbreviated as "csc," is one of the six fundamental trigonometric functions. It is particularly important because it is the reciprocal of the sine function. So, for any angle \(x\), the cosecant function is defined as:\[ \csc x = \frac{1}{\sin x} \]
This means that wherever the sine function is zero, the cosecant function tends to infinity. This phenomenon is crucial when graphing the cosecant function since it will have vertical asymptotes where the sine value is zero.

• The graph of \( \csc x \) has a series of upward and downward U-shaped curves.
• Key points of intersection occur where sine values are maximum or minimum.

These characteristics make understanding cosecant crucial when analyzing trigonometric identities involving reciprocal functions.

Cotangent Function

The cotangent function, abbreviated as "cot," serves as another pivotal trigonometric function. It is defined as the reciprocal of the tangent function:\[ \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \]
This function is undefined where the tangent function is zero, leading to vertical asymptotes in its graph.

• The cotangent graph usually appears as decreasing curves across its domain.
• It is periodic with a period of \(\pi\), signifying that the pattern repeats every \(\pi\) units.

Recognizing these properties in the context of graphing can elucidate how adjustments in trigonometric equations affect solutions. Moreover, its relation with sine and cosine functions facilitates the expression of identities, as seen with \( \csc^2 x = 1 + \cot^2 x \).

Graphing Utility

A graphing utility is an essential tool for visualizing equations and their transformations, especially when dealing with complex trigonometric identities. With graphing utilities, you can efficiently plot functions like \( y = \csc^4 x - 2 \csc^2 x + 1 \) and \( y = \cot^4 x \).

• By graphing these functions, you can visually compare whether they match over a range of \( x \)-values.
• The insights gleaned from graphical analysis are invaluable because they provide an intuitive grasp of whether two sides of an equation are identical.

Once plotted, the functions' graphical behaviors such as overlaps or disparities, help verify if given equations hold true as identities. Thus, graphing utilities streamline the process by allowing students to explore and confirm their equations visually before delving into algebraic proofs.

Algebraic Confirmation

Algebraic confirmation serves as the final and often definitive step in verifying trigonometric identities. It involves manipulating and simplifying expressions in order to prove their equivalence. For example, using known identities like\( \csc^2 x = 1 + \cot^2 x \), one can transform complex expressions.
In the given problem, substituting for the cosecant in terms of cotangent:

• Use \( \csc^2 x = 1 + \cot^2 x \) to change \( \csc^4 x - 2 \csc^2 x + 1 \) into \((\cot^2x + 1)^2 - 2(\cot^2x + 1) + 1\).
• Simplify step-by-step to show it equals \( \cot^4 x \).

This comprehensive breakdown confirms the initial graphical observations. It underscores the power of identities in proving equations beyond mere visual confirmation, thus bolstering mathematical understanding with thorough logical proofs.