Problem 62 - Precalculus: Functions and Graphs

Question

Either show that the equation \(i s\) an identity or show that the equation \(is\quad not\) an identity. $$\frac{\tan ^{2} x}{\sec x-1}=\sec x$$

Step-by-Step Solution

Answer

The equation is not an identity.

Step 1: Understand the Given Equation

We need to check if the equation \( \frac{\tan^2 x}{\sec x - 1} = \sec x \) is an identity. An identity holds for all values in its domain.

Step 2: Express Functions in Sine and Cosine

Rewrite the trigonometric functions in terms of sine and cosine. We have \( \tan x = \frac{\sin x}{\cos x} \) and \( \sec x = \frac{1}{\cos x} \). Substitute these into the equation:\[\frac{\left( \frac{\sin x}{\cos x} \right)^2}{\frac{1}{\cos x} - 1} = \frac{1}{\cos x}\]

Step 3: Simplify the Left Side

Simplify the expression on the left side by substituting the expressions for \(\tan x\) and \(\sec x\):\[\frac{\frac{\sin^2 x}{\cos^2 x}}{\frac{1 - \cos x}{\cos x}}\]The denominator becomes \(\frac{1 - \cos x}{\cos x}\). Multiply numerator and denominator by \(\cos^2 x\):\[= \frac{\sin^2 x \cdot \cos x}{1 - \cos x}\]

Step 4: Simplify Further and Compare

Simplify the numerator \(\sin^2 x \cdot \cos x\) and try to simplify both sides:\[\frac{\sin^2 x \cdot \cos x}{1 - \cos x} eq \frac{1}{\cos x}\]The expression does not simplify to match \(\sec x\), indicating a discrepancy.

Step 5: Conclude on Identity Status

Since the two sides of the equation do not simplify to equal each other for all \(x\), the equation \(\frac{\tan^2 x}{\sec x - 1} = \sec x\) is not an identity. There are values of \(x\) for which this equation does not hold, proving it is not universally true.

Key Concepts

Simplification of Trigonometric Expressions

Simplification of trigonometric expressions often involves breaking down complex equations into simpler components. This is accomplished by expressing trigonometric functions in terms of basic identities like sine and cosine. Consider the expression \( \frac{\tan^2 x}{\sec x - 1} \). A useful strategy here is to express \( \tan x \) and \( \sec x \) in terms of sine and cosine.

• \( \tan x = \frac{\sin x}{\cos x} \): Tangent is the ratio of sine to cosine.
• \( \sec x = \frac{1}{\cos x} \): Secant is the reciprocal of cosine.

By substituting these identities, complex expressions are often made more manageable. In this context, we're aiming to determine whether such transformations yield equal expressions on both sides, hinting at a deeper identity. Yet, sometimes, as in this exercise, the simplification process reveals discrepancies that indicate the expression is not an identity throughout its domain.

Sine and Cosine Functions

One of the core foundations in trigonometry is the use of sine and cosine functions. These functions are crucial as they form the basis for expressing other trigonometric identities. For instance, in our problem, tangent and secant were rewritten using sine and cosine:

• \( \tan x = \frac{\sin x}{\cos x} \)
• \( \sec x = \frac{1}{\cos x} \)

Such conversions are vital because sine and cosine are more easily manipulated mathematically due to their fundamental nature. They can lead to simplifications that make it clearer whether an equation is an identity or not. In this exercise, these expressions helped reveal the non-identity status by exposing differences between the two sides, which might not have been obvious without expressing everything in terms of these basic functions.

Equivalent Trigonometric Forms

Understanding equivalent trigonometric forms is crucial when analyzing whether two expressions are identical for every value within a certain range. Simplification often reveals these forms. For instance, when equivalent forms like \( \frac{1}{\cos x} \) for \( \sec x \) align through transformations, an identity may be possible. This concept promotes an in-depth understanding of how transformations work:

• Applying transformations converts complex expressions into simpler or comparable forms.
• Equivalent forms may appear different but are equal for all applicable values of \( x \).

In the example from the exercise, despite transformations tailored to reconcile the left with the right side of the equation, the discrepancy persisted. It showed that even if expressions can be closely dissected into relationally similar elements, they may not always line up as identities, emphasizing the skill in identifying changes and matches.