Problem 88 - A Graphical Approach to Precalculus with Limits

Question

Verify that equation is an identity. \(\frac{\cos \alpha}{\sec \alpha}+\frac{\sin \alpha}{\csc \alpha}=\sec ^{2} \alpha-\tan ^{2} \alpha\)

Step-by-Step Solution

Answer

Left and right sides simplify to 1; it's an identity.

Step 1: Rewrite in simpler terms

First, recall the definitions of the trigonometric functions involved. We have \( \sec \alpha = \frac{1}{\cos \alpha} \) and \( \csc \alpha = \frac{1}{\sin \alpha} \). Use these definitions to rewrite the left side of the equation. Therefore, \( \frac{\cos \alpha}{\sec \alpha} = \cos \alpha \times \cos \alpha = \cos^2 \alpha \) and \( \frac{\sin \alpha}{\csc \alpha} = \sin \alpha \times \sin \alpha = \sin^2 \alpha \).

Step 2: Simplify the left side

Combine results from Step 1 to simplify the left side of the equation: \( \cos^2 \alpha + \sin^2 \alpha \). According to the Pythagorean identity, this is equal to 1 since \( \cos^2 \alpha + \sin^2 \alpha = 1 \).

Step 3: Simplify the right side

Now let's simplify the right side of the equation. Recall the identities \( \sec^2 \alpha = 1 + \tan^2 \alpha \). Hence the right side \( \sec^2 \alpha - \tan^2 \alpha \) becomes \((1 + \tan^2 \alpha) - \tan^2 \alpha = 1\).

Step 4: Compare both sides

We have shown that both sides of the equation simplify to 1: left side being \(1\), and the right side being \(1\) as well. This confirms the original equation is an identity.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the fundamental relationships in trigonometry. It states that for any angle \(\alpha\), the square of the cosine plus the square of the sine is equal to one. Mathematically, this is written as:

• \(\cos^2 \alpha + \sin^2 \alpha = 1\)

This identity is crucial when dealing with trigonometric equations because it allows you to simplify expressions and verify identities. For example, if you encounter a term like \(\cos^2 \alpha + \sin^2 \alpha\), you can immediately replace it with 1, as shown in the exercise when simplifying the left side of the equation. Recognizing this identity can often be the key to unlocking more complex problems related to trigonometric functions.

Trig Simplification

Trig Simplification is the process of making trigonometric expressions easier to work with. The goal is to break down complex expressions into simpler ones using trigonometric identities, like the Pythagorean Identity. In the given problem, simplification involves rewriting complex fractions using basic trigonometric definitions. For instance:

• \(\frac{\cos \alpha}{\sec \alpha} = \cos \alpha \times \cos \alpha = \cos^2 \alpha\)
• \(\frac{\sin \alpha}{\csc \alpha} = \sin \alpha \times \sin \alpha = \sin^2 \alpha\)

By rewriting and combining terms using the Pythagorean Identity, we simplify these fractions effectively. Through such simplifications, both sides of the equation transform into more straightforward constants or expressions, making it easier to compare or solve them.

Secant and Cosecant Functions

The secant and cosecant functions are reciprocal trigonometric functions. They are defined as:

• \(\sec \alpha = \frac{1}{\cos \alpha}\)
• \(\csc \alpha = \frac{1}{\sin \alpha}\)

These functions are often involved in equations where fractions are present, as they relate directly to cosine and sine functions. Understanding how to convert secant and cosecant back to cosine and sine, as done in Step 1 of the exercise solution, is important for simplification. For example, replacing \(\sec \alpha\) and \(\csc \alpha\) helps in transforming a complex trigonometric expression into a more manageable form. Such transformations make it possible to apply fundamental identities easily and verify or solve trigonometric identities accurately.