Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-26,\) prove that each equation is an identity. $$ \sin \theta(\csc \theta-\sin \theta)=\cos ^{2} \theta $$

Step-by-Step Solution

Answer

The given equation is an identity as both sides simplify to \(\cos^2 \theta\).

Step 1: Rewrite Known Identities

Start by recalling and writing the identities for sine and cosecant: \( \csc \theta = \frac{1}{\sin \theta} \). This will help us simplify the left-hand side of the equation.

Step 2: Substitute Identity in the Left Side

Substitute \( \csc \theta \) in the equation: \( \sin \theta \left( \frac{1}{\sin \theta} - \sin \theta \right) \). This replaces \( \csc \theta \) with \( \frac{1}{\sin \theta} \).

Step 3: Simplify the Expression

Simplify the expression inside the parenthesis: \( \frac{1}{\sin \theta} - \sin \theta = \frac{1 - \sin^2 \theta}{\sin \theta} \). Use a common denominator to combine the terms.

Step 4: Use Pythagorean Identity

Recall and use the Pythagorean identity: \( 1 - \sin^2 \theta = \cos^2 \theta \). Substitute this identity into the equation: \( \sin \theta \times \frac{\cos^2 \theta}{\sin \theta} \).

Step 5: Cancel Terms

Cancel \( \sin \theta \) from the numerator and the denominator: \( \cos^2 \theta \). The equation is now simplified to \( \cos^2 \theta \).

Step 6: Verify Identity

Both sides of the original equation now equal \( \cos^2 \theta \). Therefore, this confirms the original equation is an identity.

Key Concepts

Sine and Cosecant Relationship

Understanding the relationship between sine and cosecant is fundamental in trigonometry. Sine, denoted as \( \sin \theta \), is a basic trigonometric function representing the ratio of the opposite side to the hypotenuse in a right-angled triangle.
On the other hand, cosecant, represented as \( \csc \theta \), is the reciprocal of sine. This means:

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

This reciprocal relationship is crucial when simplifying trigonometric identities as it allows us to express the cosecant function in terms of sine. By substituting \( \csc \theta \) with \( \frac{1}{\sin \theta} \), we can work with more straightforward expressions that are easier to manipulate.
Knowing these relationships helps leverage identities in different trigonometric expressions, providing a vital tool for simplification and problem-solving.

Pythagorean Identity

The Pythagorean identity is one of the most pivotal in trigonometry, acting as a cornerstone for simplifying expressions and proving identities. It is expressed as:

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

This identity derives from the Pythagorean theorem and can be rearranged in various forms to assist in different scenarios.
For example, from the basic identity, we can also deduce:

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

In this particular exercise, the form \( 1 - \sin^2 \theta = \cos^2 \theta \) is used. Recognizing these forms helps streamline the process of proving trigonometric identities, as seen in the solution step where the left-hand side simplifies to match the form of \( \cos^2 \theta \).

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves using known identities and algebraic techniques to make them easier to work with. This often includes substituting identities, combining like terms, or cancelling terms in fractions.
In the given exercise, simplification starts by substituting \( \csc \theta \) with its equivalent expression in terms of sine. After substitution, the expression becomes easier to manipulate:

• Convert \( \sin \theta (\frac{1}{\sin \theta} - \sin \theta) \) into a single fraction to simplify the terms.

Next, use the Pythagorean identity to rewrite the expression in a simpler form:

• This converts \( \frac{1 - \sin^2 \theta}{\sin \theta} \) directly into \( \frac{\cos^2 \theta}{\sin \theta} \).

Finally, cancelling \( \sin \theta \) from the numerator and denominator simplifies the expression to \( \cos^2 \theta \).
This series of steps illustrates the importance of strategic substitution and simplification in solving trigonometric expressions, making complex equations more manageable.