Problem 32 - Contemporary Precalculus

Question

State whether or not the equation is an identity. If it is an identity, prove it. $$\sec ^{4} x-\tan ^{4} x=1+2 \tan ^{2} x$$

Step-by-Step Solution

Answer

Equation: $$\sec ^{4} x - \tan ^{4} x = 1 + 2\tan ^{2} x$$ Answer: No, the given equation is not an identity.

Step 1: Convert sec and tan to sine and cosine functions

Since \(\sec x = \frac{1}{\cos x}\) and \(\tan x = \frac{\sin x}{\cos x}\), we can rewrite the given equation as: $$\left(\frac{1}{\cos ^{4} x}\right) - \left(\frac{\sin ^{4} x}{\cos ^{4} x}\right) = 1 + 2\left(\frac{\sin ^{2} x}{\cos ^{2} x}\right)$$

Step 2: Combine the terms on the left side of the equation

Both terms on the left side of the given equation have a common denominator of \(\cos ^{4} x\). We can combine them into a single fraction: $$\frac{1 - \sin ^{4} x}{\cos ^{4} x} = 1 + 2\left(\frac{\sin ^{2} x}{\cos ^{2} x}\right)$$

Step 3: Apply the Pythagorean Identity

We know that the Pythagorean Identity states that \(\sin ^{2} x + \cos ^{2} x = 1\). We can rewrite \(1 - \sin ^{4} x\) as \((1 - \sin ^{2} x)(1 + \sin ^{2} x)\): $$\frac{(1 - \sin ^{2} x)(1 + \sin ^{2} x)}{\cos ^{4} x} = 1 + 2\left(\frac{\sin ^{2} x}{\cos ^{2} x}\right)$$ Using the Pythagorean Identity, we can substitute \(\cos ^{2} x\) for \(1 - \sin ^{2} x\): $$\frac{(\cos ^{2} x)(1 + \sin ^{2} x)}{\cos ^{4} x} = 1 + 2\left(\frac{\sin ^{2} x}{\cos ^{2} x}\right)$$

Step 4: Simplify the equation

Now we can simplify the left side of the equation by canceling out one of the \(\cos ^{2} x\) terms in the numerator and denominator: $$\frac{1 + \sin ^{2} x}{\cos ^{2} x} = 1 + 2\left(\frac{\sin ^{2} x}{\cos ^{2} x}\right)$$ Now, let's subtract 1 from both sides: $$\frac{\sin ^{2} x}{\cos ^{2} x} = 2\left(\frac{\sin ^{2} x}{\cos ^{2} x}\right)$$ At this point, the equation simplifies to: $$1 = 2$$ #Conclusion# Since the left and right sides of the equation do not match, the given equation is not an identity.

Key Concepts

Secant and Tangent Functions

Understanding secant and tangent functions is crucial for solving trigonometric equations. The secant function, denoted as \( \sec x \), is the reciprocal of the cosine function. It's represented as \( \sec x = \frac{1}{\cos x} \). The tangent function, represented as \( \tan x \), is the ratio of the sine function to the cosine function and is given by \( \tan x = \frac{\sin x}{\cos x} \).

These definitions help us manipulate trigonometric expressions by converting them into terms of sine and cosine, which are more straightforward to work with in many contexts. For instance, by rewriting \( \sec^4 x \) as \( \left(\frac{1}{\cos x}\right)^4 \) and \( \tan^4 x \) as \( \left(\frac{\sin x}{\cos x}\right)^4 \), we can simplify complex expressions and make them easier to solve or verify.

Pythagorean Identity

The Pythagorean identity is one of the fundamental identities in trigonometry, linking sine and cosine. This identity is expressed as \( \sin^2 x + \cos^2 x = 1 \). It is particularly useful when verifying trigonometric equations because it allows for substitution and simplification.

In our problem, we applied the Pythagorean identity to rewrite the expression \( 1 - \sin^4 x \) as \( (1 - \sin^2 x)(1 + \sin^2 x) \). By doing so, and using the identity \( \cos^2 x = 1 - \sin^2 x \), the simplifications became more manageable. This manipulation shows how versatile the Pythagorean identity is in transforming and simplifying trigonometric expressions.

• The identity helps in simplifying expressions by eliminating squares of sine or cosine.
• It serves as a tool to connect different trigonometric functions together.

Trigonometric Equations

Trigonometric equations are equations involving trigonometric functions that must be solved for the variable angle. Solving these equations often requires the application of identities, such as the secant and tangent functions and the Pythagorean identity.

The equation given in the exercise is a trigonometric equation where the goal was to determine if it was an identity. An identity is true for all values of the variable within its domain. However, during the solving process, it was found that the equation simplifies to \( 1 = 2 \), which is a contradiction and indicates that the equation is not an identity.

When approaching trigonometric equations, it's vital to:

• Convert functions to sine and cosine for easier manipulation.
• Apply relevant identities to simplify and solve equations.
• Check each step thoroughly to ensure no algebraic errors occur.

This step-by-step approach allows for a systematic resolution of whether an equation holds as an identity, leading to a better understanding of its structure and properties.