Problem 53 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\frac{1-\cos \alpha}{\sin \alpha}=\frac{\sin \alpha}{1+\cos \alpha}$$

Step-by-Step Solution

Answer

The identity is true as both sides simplify to \( \sin \alpha \).

Step 1: Clarify the Identity to Verify

We need to verify that \( \frac{1-\cos \alpha}{\sin \alpha} = \frac{\sin \alpha}{1+\cos \alpha} \). This means showing that both sides simplify to the same expression.

Step 2: Simplify the Left Side

The left side of the identity is \( \frac{1-\cos \alpha}{\sin \alpha} \). Apply the Pythagorean identity \( 1 - \cos^2 \alpha = \sin^2 \alpha \) by rewriting the numerator: \[ 1 - \cos^2 \alpha = \sin^2 \alpha \]Now factor the numerator using the identity:\[\frac{(1 - \cos \alpha)(1 + \cos \alpha)}{\sin \alpha} = \frac{\sin^2 \alpha}{\sin \alpha} \]

Step 3: Simplify the Result

Cancel the common term \( \sin \alpha \) in the numerator and denominator:\[\frac{\sin^2 \alpha}{\sin \alpha} = \sin \alpha\]So, the left side simplifies to \( \sin \alpha \).

Step 4: Simplify the Right Side

The right side of the identity is \( \frac{\sin \alpha}{1+ \cos \alpha} \). We need to leave it in terms of \( \sin \alpha \):Immediately, it can be seen as \( \sin \alpha = \sin \alpha \times 1 \). So the right side simplifies directly to \( \sin \alpha \).

Step 5: Conclude the Verification

Both the simplified left side and right side equal \( \sin \alpha \). Therefore, the identity \( \frac{1-\cos \alpha}{\sin \alpha} = \frac{\sin \alpha}{1+\cos \alpha} \) is verified to be true.

Key Concepts

Pythagorean Identity

One fundamental concept in trigonometry is the Pythagorean Identity, which is rooted in the Pythagorean Theorem. This identity states that for any angle \( \alpha \), the sum of the square of sine and the square of cosine is always one. It is expressed as:\[\sin^2 \alpha + \cos^2 \alpha = 1\]This identity is invaluable in simplifying trigonometric expressions. It allows you to rewrite parts of an equation to reveal simpler forms or verify identities.
For instance, rearranging this identity, you can express \( \sin^2 \alpha \) as:\[\sin^2 \alpha = 1 - \cos^2 \alpha\]This re-arrangement is precisely what is employed to simplify the left side of the original exercise's identity. By recognizing and utilizing this relationship, you convert complex trigonometric expressions into easier forms, often leading to the verification or simplification of identities.

Simplifying Expressions

Simplifying expressions is a critical skill in mathematics, especially when dealing with trigonometric identities. The goal is to reduce an expression to its simplest form. This simplification often makes it easier to verify or solve trigonometric equations.
In our exercise, simplification involves applying trigonometric identities and factoring techniques. For the identity \( \frac{1-\cos \alpha}{\sin \alpha} = \frac{\sin \alpha}{1+\cos \alpha} \), the left side was simplified using the Pythagorean Identity:

• First, \( 1-\cos^2 \alpha \) is rewritten as \( \sin^2 \alpha \).
• Then, this is factored into \( (1-\cos \alpha)(1+\cos \alpha) \).

From here, you can cancel common terms in the fraction, ultimately simplifying down to a single \( \sin \alpha \). These steps illustrate how expressing components of an identity in different forms can lead to the simplification and verification of the identity. The refined expression is easier to work with and understand.

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. At its core, it focuses on trigonometric functions like sine, cosine, and tangent. These functions are fundamental in various fields such as physics, engineering, and computer science.
In this particular problem, the trigonometric functions \( \sin \alpha \) and \( \cos \alpha \) play crucial roles in forming and verifying the identity. Understanding how these functions relate and how they can be transformed or manipulated is essential for solving trigonometric problems.
By mastering these trigonometric relationships, including their associated identities like the Pythagorean Identity, you can tackle a wide array of mathematical challenges. Trigonometry provides tools and methods for modeling periodic phenomena, solving right triangles, and understanding geometry more broadly.