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Chapter 5: Problem 47

In Exercises 47-58, determine whether each equation is an identity, a conditional equation, or a contradiction. $$ \cos ^{2} x(\tan x-\sec x)(\tan x+\sec x)=1 $$

Short Answer

Expert verified
The equation is a contradiction.

Step by step solution

01

Use Trigonometric Identities

We start by simplifying the expression using trigonometric identities. Recall that \( \tan x = \frac{\sin x}{\cos x} \) and \( \sec x = \frac{1}{\cos x} \).
02

Simplify Within the Parentheses

The expression inside the parentheses \((\tan x - \sec x)(\tan x + \sec x)\) is a difference of squares that simplifies to \((\tan^2 x - \sec^2 x)\).
03

Substitute Trigonometric Values

Substitute the trigonometric definitions: \( \tan^2 x = \frac{\sin^2 x}{\cos^2 x} \) and \( \sec^2 x = \frac{1}{\cos^2 x} \). This gives \[ \tan^2 x - \sec^2 x = \frac{\sin^2 x}{\cos^2 x} - \frac{1}{\cos^2 x} = \frac{\sin^2 x - 1}{\cos^2 x} \].
04

Recognize Trigonometric Identity

Realize that \( \sin^2 x - 1 = -(1 - \sin^2 x) = -\cos^2 x \) due to the Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \).
05

Simplify Further

Replace \( \sin^2 x - 1 \) with \(-\cos^2 x \), so \( \frac{\sin^2 x - 1}{\cos^2 x} \) becomes \(-1\).
06

Substitute Back in the Original Expression

The original expression becomes \[ \cos^2 x \cdot (-1) = -\cos^2 x \].
07

Evaluate Equality with the Original Equation

Set \( -\cos^2 x = 1 \). Since \( \cos^2 x \) is always non-negative and never 1, this equality is never true.
08

Conclusion

Since the equation holds for no value of \( x \), this equation is a contradiction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares
The difference of squares is a useful algebraic identity that helps to simplify expressions. It states that for any two numbers, say "a" and "b", the difference of squares is given by: - \( (a - b)(a + b) = a^2 - b^2 \).- In trigonometry, this is particularly helpful. In our exercise, we applied the difference of squares to the expression \((\tan x - \sec x)(\tan x + \sec x)\).- This was simplified to \( \tan^2 x - \sec^2 x \).Understanding this identity can simplify many problems and makes it easier to manipulate trigonometric expressions for further simplification.
Conditional Equation
Conditional equations are equations that are true only for specific values of the variable. They are not true in a general sense.- For example, the equation \( x^2 = 4 \) is conditional because it holds true only when \( x = 2 \) or \( x = -2 \).In our trigonometric problem, however, we found that the equation does not hold true for any value of \( x \). Thus, it is not a conditional equation.- Instead, it falls under another category, which we will explore next.
Contradiction
A contradiction in equations is when both sides of the equation can never be equal, no matter what values you insert for the variables.- In our exercise with the equation \( \cos^2 x \cdot (-1) = 1 \), we found that it is indeed a contradiction.- This is because \( \cos^2 x \) represents a range of values between 0 and 1 (including both), and multiplying by -1 can never equal 1.Understanding when an equation is a contradiction helps identify unrealistic solutions, saving time and effort in seeking possible real-world applications or interpretations.
Pythagorean Identity
The Pythagorean identity is one of the basic identities in trigonometry. It is expressed as:- \( \sin^2 x + \cos^2 x = 1 \).- This identity allows us to express one trigonometric function in terms of another.In solving the given equation, we applied the Pythagorean identity. To break down the expression \( \sin^2 x - 1 \), we recognized it as \( -\cos^2 x \) using the identity.- This step is crucial, as it assists in the simplification process and combines trigonometric expressions into a more manageable form.By mastering the Pythagorean identity, solving complex trigonometric problems becomes more intuitive and straightforward.

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Most popular questions from this chapter

A bicycle ramp is made so that it can easily be raised and lowered for different levels of competition. For the advance division, the angle formed by the ramp and the ground is \(\theta\) such that \(\tan \theta\), the steepness of the ramp, is \(\frac{5}{3}\). For the novice division, the angle \(\theta\) is cut in half to lower the ramp. What is the steepness of the ramp for angle \(\frac{\theta}{2}\) ?

Find the exact value of \(\tan 15^{\circ}\) in two ways, using sum and difference identities and half-angle identities; then show that they are equal.

For Exercises 69-72, refer to the following: We cannot prove that an equation is an identity using technology, but we can use technology as a first step to see whether or not the equation seems to be an identity. Using a graphing calculator, plot \(Y_{1}=(2 x)-\frac{(2 x)^{3}}{3 !}+\frac{(2 x)^{5}}{5 !}\) and \(Y_{2}=\sin (2 x)\) for \(x\) ranging \([-1,1]\). Is \(Y_{1}\) a good approximation to \(Y_{2}\) ?

An auger used to deliver grain to a storage bin can be raised and lowered, thus allowing for different size bins. Let \(\alpha\) be the angle formed by the auger and the ground for bin \(\mathrm{A}\) such that \(\sin \alpha=\frac{40}{41}\). The angle formed by the auger and the ground for bin B is half of \(\alpha\). If the height \(h\), in feet, of a bin can be found using the formula \(h=75 \sin \theta\), where \(\theta\) is the angle formed by the ground and the auger, find the height of bin \(\mathrm{B}\).

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