Problem 28 - Algebra 2

Question

Verify that each of the following is an identity. $$ \frac{1-\cos \theta}{\sin \theta}=\frac{\sin \theta}{1+\cos \theta} $$

Step-by-Step Solution

Answer

The equation is an identity because both sides simplify to \( \frac{\sin \theta}{1+\cos \theta} \).

Step 1: Understand the Problem

We need to verify if \( \frac{1-\cos \theta}{\sin \theta} = \frac{\sin \theta}{1+\cos \theta} \) is an identity. This means showing that both sides of the equation are equal for all permissible values of \( \theta \).

Step 2: Simplify Left Side

The left side is \( \frac{1-\cos \theta}{\sin \theta} \). To simplify, we multiply the numerator and the denominator by \( 1+\cos \theta \) to get: \[ \frac{(1-\cos \theta)(1+\cos \theta)}{\sin \theta (1+\cos \theta)}. \] The numerator becomes \( 1 - \cos^2 \theta \).

Step 3: Apply Pythagorean Identity

Using the Pythagorean identity \( 1 - \cos^2 \theta = \sin^2 \theta \), we have \[ \frac{\sin^2 \theta}{\sin \theta (1+\cos \theta)}. \] We can then simplify further by cancelling \( \sin \theta \) in the numerator and denominator.

Step 4: Simplify Further

After cancelling \( \sin \theta \), the left side simplifies to \( \frac{\sin \theta}{1+\cos \theta} \), which is the same as the right side.

Step 5: Verify Equality

Since both sides simplify to the same expression, \( \frac{\sin \theta}{1+\cos \theta} \), this confirms that the given equation is an identity.

Key Concepts

Pythagorean Identity

The Pythagorean identity is a crucial concept in trigonometry and plays a significant role in solving and simplifying trigonometric equations. It is expressed as:

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

This identity highlights the intrinsic relationship between the sine and cosine of any angle, \( \theta \), on the unit circle. By rearranging it, you get two other useful forms:

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

These rearrangements can be incredibly useful in simplifying complex trigonometric expressions. For example, in our original exercise, the expression \( 1 - \cos^2 \theta \) is replaced by \( \sin^2 \theta \) to aid simplification. This identity holds for all angles \( \theta \) and is a powerful tool in verifying identities. The Pythagorean identity provides a way to manipulate expressions into more manageable forms, often making it easier to demonstrate equivalencies.

Trigonometric Simplification

Trigonometric simplification involves reducing a trigonometric expression to its simplest form while maintaining equality. This is often done using fundamental identities, such as the Pythagorean identity, to transform the equation into a form that's easier to work with. In the given exercise, both sides of the equation use the properties of algebraic manipulation and trigonometric identities to reach a point where they appear identical.

• Multiplication and division: By multiplying the numerator and the denominator by conjugates or simplifying factors.
• Cancelling terms: Remove common terms from the numerator and the denominator to simplify the expression.

In our problem, the left-hand side of the equation \( \frac{1-\cos \theta}{\sin \theta} \), was multiplied by \( 1+\cos \theta \). This step aids in applying the Pythagorean identity effectively and leads to further simplification.Additionally, after applying\( \sin^2 \theta = 1 - \cos^2 \theta \), the term \( \sin \theta \) was cancelled out on both sides of the fraction, simplifying it to resemble the right-hand side of the original equation.

Verifying Identities

Verifying identities is a systematic process of proving that both sides of a trigonometric equation are equivalent for all their permissible values. This process involves:

• Transforming both sides of the equation into a common expression through substitutions and simplifications.
• Utilizing known identities (like the Pythagorean identity) to make expressions apparent.
• Simplifying each part step by step to reach a common outcome.

In our exercise, the process involved showing that transforming the left side of the equation \( \frac{1-\cos \theta}{\sin \theta} \) simplifies in such a way that it matches the right side \( \frac{\sin \theta}{1+\cos \theta} \).During the verification process, it is crucial to apply trigonometric identities accurately, and to reevaluate each step to ensure no mistakes are made.Ultimately, verifying identities strengthens a student's grasp of trigonometric principles and helps hone problem-solving skills. A successfully verified identity serves as a powerful testament to the interconnected nature of trigonometric expressions.