Problem 37 - A Graphical Approach to Precalculus with Limits

Question

Write expression as a single trigonometric function or a power of a trigonometric function. (You may wish to use a graph to support your result.) $$\frac{\sin ^{2} x}{\cos ^{2} x}+\sin x \csc x$$

Step-by-Step Solution

Answer

The expression simplifies to \( \sec^2 x \).

Step 1: Simplify Each Term Separately

We start by simplifying the terms in the given expression separately. The first term is \( \frac{\sin^2 x}{\cos^2 x} \), which can be rewritten using the tangent identity as \( \tan^2 x \). The second term is \( \sin x \csc x \), which simplifies to \( 1 \), because \( \csc x = \frac{1}{\sin x} \), and \( \sin x \cdot \csc x = \sin x \cdot \frac{1}{\sin x} = 1 \).

Step 2: Combine the Simplified Terms

Now that we have simplified each term, combine the results. The expression becomes \( \tan^2 x + 1 \).

Step 3: Express as a Single Trigonometric Function

Recall the Pythagorean identity \( \tan^2 x + 1 = \sec^2 x \). Substitute this identity into the expression: \( \sec^2 x \).

Step 4: Finalize the Expression

The expression is now simplified to a single trigonometric function, \( \sec^2 x \), which is the result.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental relationship in trigonometry that connects the squares of sine and cosine functions. It states that for any angle \( x \), \( \sin^2 x + \cos^2 x = 1 \). This identity provides a valuable tool for simplifying and transforming trigonometric expressions.In the context of the given problem, another variation of the Pythagorean Identity is utilized: \( \tan^2 x + 1 = \sec^2 x \). This variation arises from the basic identity and is derived by dividing every term by \( \cos^2 x \), considering \( \tan x = \frac{\sin x}{\cos x} \) and \( \sec x = \frac{1}{\cos x} \).
This identity is crucial when simplifying expressions like \( \tan^2 x + 1 \) because it allows us to replace it with a single, more convenient trigonometric function, \( \sec^2 x \). As we observe these transformations, it becomes clear how the various trigonometric identities work together, highlighting patterns and relationships between different functions.
To master these concepts, it can be helpful to visualize the identities by sketching graphs of trigonometric functions and observing intersections or particular points where the identities hold true.

Tangent and Secant Functions

The tangent and secant functions

• Tangent function: Given as \( \tan x = \frac{\sin x}{\cos x} \), represents the slope of a triangle formed with the unit circle. It is undefined at positions where \( \cos x = 0 \).
• Secant function: Denoted as \( \sec x = \frac{1}{\cos x} \), is the reciprocal of the cosine function. It is undefined where \( \cos x = 0 \), similar to the vertical asymptotes of tangent.

Both the tangent and secant functions originate from the sine and cosine functions, which are based on the unit circle. By breaking down the given expression \( \frac{\sin^2 x}{\cos^2 x} + \sin x \csc x \), we encounter the tangent function directly when \( \tan^2 x \) is derived from \( \frac{\sin^2 x}{\cos^2 x} \). The terms translate nicely because of their strong relationship with their foundational functions: sine and cosine.
Ultimately, replacing \( \tan^2 x + 1 \) with \( \sec^2 x \) in the solution is a demonstration of how interconnected tangent and secant are due to the Pythagorean Identity, emphasizing the utility of understanding these functions in problem-solving.

Trigonometric Simplification

Trigonometric simplification involves breaking down complex trigonometric expressions into simpler forms. This often requires leveraging trigonometric identities to transform the expression efficiently. Consider the given exercise: you start with \( \frac{\sin^2 x}{\cos^2 x} + \sin x \csc x \).

• Simplifying Each Term: Begin by simplifying each part of the expression separately. Dissect \( \frac{\sin^2 x}{\cos^2 x} \) to get \( \tan^2 x \) and combine the other part \( \sin x \csc x \) to result in \( 1 \), as \( \csc x \) is the reciprocal of sine.
• Combining Terms Together: Add the simplified expressions: \( \tan^2 x + 1 \).
• Using Identities for Further Simplification: Apply the Pythagorean Identity, transforming \( \tan^2 x + 1 \) into \( \sec^2 x \).

The key to simplifying trigonometric expressions is understanding and applying these identities correctly. Consistent practice with various identities helps uncover the most effective ways to transform such expressions. As these identities coordinate our transformation tactics, they reveal not just correct outcomes, but also a deeper understanding of the inherent structure within trigonometric functions.