Problem 79 - A Graphical Approach to Precalculus with Limits

Question

Verify that equation is an identity. \(\frac{1-\sin ^{2} \beta}{\cos \beta}=\cos \beta\)

Step-by-Step Solution

Answer

The equation is an identity.

Step 1: Identify the Trigonometric Identity

The equation given is \( \frac{1-\sin^2 \beta}{\cos \beta} = \cos \beta \). To verify whether this is an identity, we need to simplify the left-hand side (LHS) and see if it equals the right-hand side (RHS).

Step 2: Simplify the Left-Han Side Using Trigonometric Identities

Recall the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \). This can be rearranged to \( 1-\sin^2 \theta = \cos^2 \theta \). Using this, replace \( 1-\sin^2 \beta \) with \( \cos^2 \beta \) in the LHS:\[ \frac{1-\sin^2 \beta}{\cos \beta} = \frac{\cos^2 \beta}{\cos \beta} \].

Step 3: Simplify the Fraction

Now, simplify \( \frac{\cos^2 \beta}{\cos \beta} \). This cancels out to \( \cos \beta \) because \( \cos^2 \beta \) divided by \( \cos \beta \) is \( \cos \beta \).

Step 4: Compare the Simplified LHS to the RHS

After simplification, the left-hand side \( \cos \beta \) is equal to the right-hand side \( \cos \beta \). Thus, both sides are equal, which verifies that the equation is an identity.

Key Concepts

Understanding the Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry and is represented as:

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

This identity always holds true for any angle \( \theta \). It connects the sine and cosine functions in a simple and powerful way. You can rearrange this identity to express either \( \sin^2 \theta \) or \( \cos^2 \theta \) in terms of the other. For example,

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

In our exercise, we used \( 1 - \sin^2 \beta = \cos^2 \beta \) to simplify the expression \( \frac{1 - \sin^2 \beta}{\cos \beta} \). This transformation is essential in many trigonometric problems where you need to simplify or verify expressions.

Simplifying Expressions using Identities

Simplification is a key skill in mathematics that helps make complex expressions easier to understand and work with. In trigonometry, the use of identities like the Pythagorean identity is crucial for simplifying expressions.
Our exercise began with the expression \( \frac{1 - \sin^2 \beta}{\cos \beta} \). By recognizing that \( 1 - \sin^2 \beta \) can be replaced with \( \cos^2 \beta \) using the Pythagorean identity, we transformed this into \( \frac{\cos^2 \beta}{\cos \beta} \).
Simplifying a fraction can often involve canceling out equivalent terms. Here, \( \cos^2 \beta \) divided by \( \cos \beta \) simplifies to \( \cos \beta \).

• First, identify the identity that will simplify the expression.
• Apply the identity to transform the expression.
• Cancel any common terms to simplify further.

Through these steps, we've reduced a potentially complex expression to its simplest form.

Verifying Trigonometric Identities

Verifying an identity in trigonometry means showing that two sides of an equation are equivalent by transforming one side to match the other using trigonometric identities.
In the exercise, we began with the equation \( \frac{1-\sin^2 \beta}{\cos \beta} = \cos \beta \). Our goal was to demonstrate that the left-hand side (LHS) simplifies to become the right-hand side (RHS).

• Start by focusing on one side of the equation, usually the more complex one.
• Use known identities such as the Pythagorean identity to simplify expressions.
• Step-by-step simplification helps in verifying if the equation is indeed an identity.

After simplifying the LHS to \( \cos \beta \), we saw that it is equal to the RHS. This equality confirms that the given equation is a trigonometric identity. Understanding this process is essential for solving more advanced problems and verifying similar identities in trigonometry.