Problem 84 - Algebra 2

Question

Simplify each expression. $$ \csc ^{2} \theta\left(1-\cos ^{2} \theta\right) $$

Step-by-Step Solution

Answer

The simplified form of the expression is 1.

Step 1: Use Pythagorean identity

Replace \(1-\cos^2\theta\) with \(\sin^2\theta\), using the Pythagorean identity. This transforms the expression into \(\csc^2\theta \cdot \sin^2\theta\).

Step 2: Replace cosecant

We know that \(\csc\theta = \frac{1}{\sin\theta}\), so \(\csc^2\theta\) can be replaced by \(\frac{1}{\sin^2\theta}\). So the equation becomes \(\frac{1}{\sin^2\theta} \cdot \sin^2\theta\).

Step 3: Simplify the expression

\(\frac{1}{\sin^2\theta} \cdot \sin^2\theta\) simplifies to 1, making that the final expression.

Key Concepts

Understanding the Cosecant Function

The cosecant function, often abbreviated as \( \csc \theta \), is a fundamental trigonometric function closely tied to the sine function. It is defined as the reciprocal of the sine function:

• \( \csc \theta = \frac{1}{\sin \theta} \)
Given this relationship, the cosecant function helps in expressing and analyzing angles in different contexts. It's crucial for students to remember that unlike sine, which takes values between -1 and 1, the cosecant takes values outside this range. This is because it's undefined when the sine of an angle is zero, thus creating vertical asymptotes at those points.
In trigonometric expressions and equations, understanding how to manipulate the cosecant function can simplify complex expressions, improve problem-solving efficiency, and aid in proving identities.

Exploring the Pythagorean Identity

The Pythagorean identity is one of the cornerstone trigonometric identities. It provides a relationship between sine, cosine, and the Pythagorean theorem. The identity is given by:

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

This simple yet powerful identity can be rearranged to express \( \sin^2 \theta \) in terms of \( \cos^2 \theta \), which is precisely what was done in the problem solution:

• \( \sin^2 \theta = 1 - \cos^2 \theta \)

Using this identity allows us to substitute \( 1 - \cos^2 \theta \) directly with \( \sin^2 \theta \), significantly simplifying trigonometric expressions. This is particularly useful when dealing with expressions involving squares of sine or cosine, as it enables conversion and simplification, facilitating easier manipulation and solution.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions can make solving equations or proving identities more straightforward. In the given exercise, simplifying started by employing the Pythagorean identity. By replacing \( 1 - \cos^2 \theta \) with \( \sin^2 \theta \), we set the ground for the simplification of the expression \( \csc^2 \theta \cdot \sin^2 \theta \).
The next key step was replacing \( \csc^2 \theta \) with its reciprocal identity, \( \frac{1}{\sin^2 \theta} \). This substitution is crucial because it turns the expression into a simple multiplication problem:

• \( \frac{1}{\sin^2 \theta} \cdot \sin^2 \theta = 1 \)

The sine terms in numerator and denominator cancel each other out, leading to an elegantly simplified result of 1. This approach of systematically using identities and substitutions to simplify expressions makes the task of solving trigonometric problems more manageable and less prone to error.