Problem 90 - A Graphical Approach to Precalculus with Limits

Question

Verify that each equation is an identity by using any of the identities introduced in the first three sections of this chapter. $$\csc \theta \cos ^{2} \theta+\sin \theta=\csc \theta$$

Step-by-Step Solution

Answer

The given equation is an identity since both sides simplify to \( \csc \theta \).

Step 1: Understand the identity to be verified

The given equation to verify as an identity is \( \csc \theta \cos^2 \theta + \sin \theta = \csc \theta \). We need to verify if both sides of this equation are indeed identical.

Step 2: Start with simplifying the left side

We'll start by working on the left side of the equation: \( \csc \theta \cos^2 \theta + \sin \theta \). Recall that \( \csc \theta = \frac{1}{\sin \theta} \), so substitute that in: \( \frac{1}{\sin \theta} \cos^2 \theta + \sin \theta \).

Step 3: Simplify the expression

The expression \( \frac{1}{\sin \theta} \cos^2 \theta + \sin \theta \) can be written as \( \frac{\cos^2 \theta}{\sin \theta} + \sin \theta \). This needs to be combined into a single fraction.

Step 4: Combine into a single fraction

Combine \( \frac{\cos^2 \theta}{\sin \theta} + \sin \theta \) into a single fraction with a common denominator \( \sin \theta \), obtaining \( \frac{\cos^2 \theta + \sin^2 \theta}{\sin \theta} \).

Step 5: Use Pythagorean Identity

Recognize that \( \cos^2 \theta + \sin^2 \theta \) is a Pythagorean identity equal to 1. Replace it to get \( \frac{1}{\sin \theta} \).

Step 6: Verify equivalence to the right side

Notice that \( \frac{1}{\sin \theta} \) is equal to \( \csc \theta \), thus confirming that the left side simplifies to match the right side of the equation.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is an important trigonometric function used in many mathematical identities. It is the reciprocal of the sine function. This means that \( \csc \theta = \frac{1}{\sin \theta} \). Understanding this relationship lies at the heart of many simplification techniques in trigonometry.
A key point when working with the cosecant function is to know how it behaves and transforms other trigonometric identities. By expressing \( \csc \theta \) in terms of other functions like sine, it's easier to work with and simplify expressions. This is particularly helpful when verifying trigonometric identities, as it allows you to change forms and identify common patterns.

Pythagorean Identity

One of the fundamental relationships in trigonometry is the Pythagorean identity. It states that \( \cos^2 \theta + \sin^2 \theta = 1 \). This identity is often used to simplify complex trigonometric expressions and equations.
In the context of verifying the given identity, the Pythagorean identity allows us to replace the sum \( \cos^2 \theta + \sin^2 \theta \) with 1. This step is crucial in moving from a more complex expression to something simpler, and ultimately, demonstrating that both sides of the equation are equal. The Pythagorean identity is versatile and foundational; that’s why understanding this identity gives you a powerful tool for dealing with various trigonometric problems.

Simplifying Expressions

Simplifying trigonometric expressions is a common task when dealing with identities. It involves using known identities and relationships to transform an expression into a simpler or more familiar form.
Start by identifying parts of the expression that can be substituted using identities you know. In our exercise, simplifying was achieved by using the relationship \( \csc \theta = \frac{1}{\sin \theta} \) and the Pythagorean identity. The goal is to express the left side of the equation in a form that is evidently equal to the right side. This typically involves finding a common denominator or substituting equivalent expressions. By breaking down each part systematically, you can reveal the underlying equality of both sides, verifying the identity in a step-by-step manner.