Problem 70 - Precalculus: A Concise Course

Question

Use a graphing utility to graph the two equations in the same viewing window. Use the graphs to determine whether the expressions are equivalent. Verify the results algebraically. $$ y_{1}=\tan x \cot ^{2} x, \quad y_{2}=\cot x $$

Step-by-Step Solution

Answer

The expressions \(y_{1}= \tan x \cdot \cot^{2} x\) and \(y_{2} = \cot x\) are equivalent.

Step 1: Graphing the equations

Use a graphing utility to graph \(y_{1} = \tan x \cdot \cot^{2}x\) and \(y_{2} = \cot x\) on the same viewing window. Note the behavior of the graphs, specifically any similarities and differences.

Step 2: Comparison of Graphs

Based on the graphs, make an initial comparison whether \(y_{1}\) and \(y_{2}\) exhibit similar behavior or not. If the graphs coincide for all \(x\), then the expressions are visually equivalent.

Step 3: Algebraic Verification

To algebraically verify the results, express both \(y_{1}\) and \(y_{2}\) in terms of sine and cosine, since \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \). Hence, \(y_{1} = \frac{\sin x}{\cos x} \cdot \left(\frac{\cos x}{\sin x}\right)^{2} \) and \( y_{2} = \frac{\cos x}{\sin x} \).

Step 4: Reduction of Algebraic Expressions

Simplify \(y_{1}\) by multiplying out and cancelling common terms. This should be done by multiplying \(\frac{\sin x}{\cos x}\) by \(\left(\frac{\cos^{2} x}{\sin^{2} x}\right)\) to get \(\frac{\cos x}{\sin x}\), which is equivalent to \(y_{2}\).

Step 5: Final Verification

Having simplified \(y_{1}\) to \(y_{2}\), it is confirmed that \(y_{1}\) and \(y_{2}\) are indeed equivalent.

Key Concepts

Graphing Utility

One essential tool for studying mathematical functions, especially trigonometric ones, is a graphing utility. These are often digital tools or software that allow students to visualize equations by plotting their graphs over a set of points. For the given problem, a graphing utility is used to plot the functions

• \(y_{1} = \tan x \cdot \cot^{2}x\)
• \(y_{2} = \cot x\)

Understanding the graphs can be as crucial as understanding the equations themselves. When plotted correctly, you can observe where the graphs intersect, diverge, or if they mirror each other over a range of \(x\) values.
Using the graphing utility, the goal is to determine if these functions overlap entirely across the same window. If they do not, the expressions are not equivalent in those regions. This step gives a visual understanding before delving into the algebraic verification.

Trigonometric Functions

Trigonometric functions are foundational in understanding wave patterns and circular motion often used in physics and engineering. In our problem, the two equations involve the tangent and cotangent functions.

• \( \tan x \) represents the ratio of the opposite to adjacent side in a right triangle, defined as \( \tan x = \frac{\sin x}{\cos x} \).
• \( \cot x \) is the reciprocal of tangent, given by \( \cot x = \frac{\cos x}{\sin x} \).

Graphing these accurately shows periodic behavior typical of trigonometric functions. The frequencies and amplitude can be compared visually. As we try to determine equivalency, noting how these functions simplify and behave graphically will help identify any similarities. Both these functions have periodic \(x\)-values where they might be undefined, influencing their graphs.

Algebraic Verification

Algebraic verification involves breaking down the trigonometric identities into their sine and cosine components and simplifying them. This process often reveals underlying equivalencies not immediately obvious from the equations' forms.
For \(y_{1} = \tan x \cdot \cot^{2} x\), substituting gives:\[y_{1} = \frac{\sin x}{\cos x} \cdot \left(\frac{\cos x}{\sin x}\right)^2 \]For \(y_{2} = \cot x\), substituting gives:\[y_{2} = \frac{\cos x}{\sin x} \]By simplifying \(y_{1}\):\[y_{1} = \frac{\sin x}{\cos x} \cdot \frac{\cos^2 x}{\sin^2 x} = \frac{\cos x}{\sin x}\]It's clear that this reduces to \(y_{2}\), confirming the equivalency of \(y_{1}\) and \(y_{2}\). This algebraic step secures what the graph visually suggested - that these expressions are equivalent. By understanding these transformations, you gain clarity on how trigonometric identities interrelate.