Problem 36 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Substitute the Pythagorean identity

In the expression \(\csc ^{2} \theta (1-\cos ^{2} \theta)\), substitute the Pythagorean identity \(1 - \cos^2\theta =\sin^2\theta\). This will result in \(\csc ^{2} \theta \sin^2\theta \).

Step 2: Simplify the expression

The expression \(\csc ^{2} \theta \sin^2\theta\) can be simplified using the definition of cosecant, which is the reciprocal of sine, or \(\csc\theta = 1/\sin\theta\). Thus the expression becomes \( (1/\sin\theta)^{2} \sin^2\theta\), or \(1\).

Key Concepts

Cosecant

Cosecant is one of the six fundamental trigonometric functions. It is particularly useful in certain mathematical expressions, especially those involving the reciprocal of sine. When we talk about cosecant of an angle \( \theta \), denoted as \( \csc \theta \), it is defined as the reciprocal of the sine function:

• \( \csc \theta = \frac{1}{\sin \theta} \)

Understanding the concept of cosecant helps simplify trigonometric expressions, especially when they involve products or fractions containing sine.It is important to remember that cosecant is undefined when sine is zero, because division by zero is not possible in mathematics.Cosecant is commonly used in simplifying complex trigonometric expressions because it transforms multiplicative sine parts into divisions, as evidenced by our original exercise.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry that relates the square of sine and cosine functions. This identity is:

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

This equation can be rearranged to express one function in terms of the other. For example, from the identity, you can derive:

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

In the original problem, the Pythagorean identity is used to replace \( 1 - \cos^2\theta \) with \( \sin^2\theta \), which simplifies the expression involving cosecant and sine. This identity is extremely useful in simplifying expressions and proving other identities in trigonometry.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the involved variables within their domains. These identities are crucial tools in simplifying trigonometric expressions and solving trigonometric equations.Some of the most commonly used trigonometric identities include:

• Pythagorean Identities: \( \sin^2\theta + \cos^2\theta = 1 \)
• Reciprocal Identities: \( \csc\theta = \frac{1}{\sin\theta}, \sec\theta = \frac{1}{\cos\theta}, \cot\theta = \frac{1}{\tan\theta} \)

Using these identities, you can transform complex trigonometric expressions into simpler forms, making them easier to understand and compute. They are essential tools in the toolbox of anyone working with trigonometric expressions or solving trigonometric problems.

Expression Simplification

Expression simplification in trigonometry involves manipulating an equation or expression into its simplest form using various identities and the relationships between the trigonometric functions.In the provided exercise, simplification was achieved by:

• Substituting \( 1 - \cos^2\theta \) with \( \sin^2\theta \) using the Pythagorean identity.
• Rewriting \( \csc^2\theta \sin^2\theta \) using the definition of cosecant, which results in simplification to \( 1 \).

These steps reduce the complexity of the expression, making it easier to interpret and work with, demonstrating that with strategic use of identities and reciprocal relationships, seemingly complex problems can become quite simple. Simplification is a powerful tool in mathematics, aiding in both comprehension and computation.