Problem 53 - 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. \( (1 + \cot^2 x) (\cos^2 x) = \cot^2 x \)

Step-by-Step Solution

Answer

The equation \( (1 + \cot^2 x) (\cos^2 x) = \cot^2 x \) is an identity

Step 1 - Graphical representation

Using a graphing utility, plot the two sides of the equation i.e., \(y1= (1 + \cot^2 x) (\cos^2 x)\) and \(y2= \cot^2 x\). The graph will show two overlapping curves, indicating that the equation could be an identity.

Step 2 - Tabular representation

Use a graphing utility's table feature, and check the values of the y1 and y2 for every x. If at every x, y1 and y2 have the same values, this provides additional evidence that the equation could be an identity.

Step 3 - Algebraic verification

To algebraically confirm the identity: \n1. Rewrite \(\cot^2 x\) as \(\frac{\cos^2 x}{\sin^2 x}\) \n2. Now, expand the left side of the equation to \((\cos^2 x) + (\frac{\cos^4 x}{\sin^2 x})\) \n3. Knowing that \(\sin^2 x+ \cos^2 x = 1\), replace \(\cos^2 x\) on the left side with \((1 - \sin^2 x)\).\n4. The equation then becomes \(1- \sin^2 x + \frac{(1- \sin^2 x)^2}{\sin^2 x}.\) Divide everything by \(\sin^2 x\) to get \(\frac{\cos^2 x}{\sin^2 x} = \cot^2 x\).\n5. Therefore, the equation is an identity.

Key Concepts

Graphing Utility Usage

Graphing utilities are powerful tools for visualizing mathematical concepts, and their usage is particularly effective in the study of trigonometry. When faced with a complex trigonometric equation or identity, graphing the functions involved can provide an immediate visual confirmation of whether the two sides of the equation match.

For the given exercise, a student would graph the functions \((1 + \cot^2 x)(\cos^2 x)\) and \(\cot^2 x\) separately on a graphing calculator or software. If the two graphs coincide throughout the range of \(x\), it suggests that the two expressions are equivalent for all values of \(x\), supporting the hypothesis that the equation is an identity. This visual check is often an efficient first step before proceeding to algebraic verification.

Trigonometric Equation Identity Verification

Verifying trigonometric identities is a fundamental aspect of trigonometry. An identity is an equation that is true for all values within the domain of the variables involved. To verify whether a given trigonometric equation is an identity, graphing utilities can again come in handy, especially with the function to do a table of values.

Using the table feature, students can list the output values for both expressions at various angles, comparing them side-by-side. If they match across a representative range of angles, this increases confidence in the equation being an identity. However, while these graphical methods are helpful, algebraic manipulation remains the definitive method to prove that an equation is indeed a trigonometric identity.

Algebraic Manipulation in Trigonometry

Trigonometry is rich with identities, and algebraic manipulation is often needed to simplify expressions or prove identities. It involves rearranging the equation using algebraic properties and trigonometric identities, such as \(\sin^2 x + \cos^2 x = 1\) or the relationships between cotangent, sine, and cosine.

In the provided exercise, the identity was confirmed through a series of algebraic steps that leveraged known trigonometric identities to simplify and transform one expression into another, ultimately showing that both sides of the equation are equivalent. Mastering these skills is crucial for students as algebraic manipulation provides the necessary proof that confirms the identity beyond just a visual or tabular match.

Cotangent and Cosine Relationship

Understanding the relationship between various trigonometric functions is key to working through identities. In particular, cotangent, which is the reciprocal of tangent, can be expressed in terms of cosine and sine as \(\cot x = \frac{\cos x}{\sin x}\). Given this relationship, \(\cot^2 x\) is therefore \(\frac{\cos^2 x}{\sin^2 x}\).

When dealing with trigonometric equations, recognizing these relationships allows for the substitution and simplification of terms, as seen in the exercise solution where \(\cot^2 x\) is replaced by \(\frac{\cos^2 x}{\sin^2 x}\). This step is vital in the algebraic manipulation process to reduce the equation and eventually establish the identity.