Problem 62 - Precalculus with Limits

Question

In Exercises 61-64, verify the identity. \( \sec^4 x \tan^2 x = (\tan^2 x + \tan^4 x) \sec^2 x \)

Step-by-Step Solution

Answer

Yes, the given trigonometric identity holds when \( \cos(x) \neq 0 \)

Step 1: Express in terms of sine and cosine

Start by expressing the secants and tangents of the equation in terms of sine and cosine. This allows for direct comparability between both sides of the equation. So, this gives: \[ (\frac{1}{\cos^4 x})(\frac{\sin^2 x}{\cos^2 x}) = ((\frac{\sin^2 x}{\cos^2 x})^2 + (\frac{\sin^2 x}{\cos^2 x})^4) (\frac{1}{\cos^2 x}) \]

Step 2: Simplify the equation

Next, simplify the equation to make the comparisons more straightforward. This gives: \[ \frac{\sin^2 x}{\cos^6 x} = \frac{\sin^4 x}{\cos^4 x} + \frac{\sin^6 x}{\cos^6 x} \]

Step 3: Check the match

Now at this step, it can be observed that both sides of the equation match if and only if \[ \frac{\sin^6 x}{\cos^6 x} = 0 \] This holds for all values of \(x\) except for those where \(\cos(x) = 0\). Hence, the given identity holds for all \(x\) such that \(\cos(x) \neq 0\). This completes the verification.

Key Concepts

Secant Function

The secant function, denoted as \( \sec(x) \), is one of the trigonometric functions often encountered in mathematical exercises involving angle measures. It is defined as the reciprocal of the cosine function, meaning that \( \sec(x) = \frac{1}{\cos(x)} \). This means that whenever the cosine of an angle is known, the secant can be quickly found by taking its reciprocal.

Key features of the secant function include:

• It is undefined for angles where \( \cos(x) = 0 \), such as \( x = \frac{\pi}{2}, \frac{3\pi}{2}, \) and so on, leading to vertical asymptotes in its graph.
• The graph of the secant function is periodic with a period of \( 2\pi \).
• It is always greater than or equal to 1 or less than or equal to -1, since \( \sec(x) \) is the reciprocal of values ranging from -1 to 1.

Understanding the secant function is essential in trigonometric identities because it often simplifies computations by converting cosine-based equations into their reciprocal form.

Tangent Function

The tangent function, represented as \( \tan(x) \), is another elementary trigonometric function. It is defined as the ratio of the sine function to the cosine function, or \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). This ratio gives the slope of a line that is tangent to the angle in a unit circle context.

Important points about the tangent function:

• \( \tan(x) \) is undefined at angles where \( \cos(x) = 0 \); these are points where the sine function crosses an axis while cosine is zero, leading to vertical asymptotes.
• The tangent function has a periodicity of \( \pi \), meaning it repeats every \( \pi \) radians.
• The graph of the tangent function crosses the origin and goes to infinity at half-period intervals, creating a wave-like oscillation.

In the context of verifying identities, expressing terms using the tangent function can simplify expressions involving division between sine and cosine, as seen in the original exercise.

Sine and Cosine Transformation

Sine and cosine transformations are powerful tools used when working with trigonometric identities. Converting expressions involving secant and tangent into sine and cosine is a typical strategy for simplifying or verifying identities, as these two foundational functions allow for more straightforward comparisons.

Here are essential insights about these transformations:

• Sine, \( \sin(x) \), and cosine, \( \cos(x) \), form the basis for defining other trigonometric functions, including secant and tangent.
• Transformations such as \( \sec(x) = \frac{1}{\cos(x)} \) and \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) help express complex trigonometric functions in terms of the more fundamental sine and cosine.
• In equations, expressing all components in terms of \( \sin(x) \) and \( \cos(x) \) is often the initial step to simplify comparison, as seen in the solution process where the original expression is rewritten using these foundational functions.

Understanding and applying sine and cosine transformations are crucial for effectively tackling trigonometric problems, ensuring simplification and successful identity verification.