Problem 51 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The identity is verified: both sides simplify to \(\frac{\sin \alpha}{1+\cos \alpha}\).

Step 1: Understand the Problem

The problem requires verifying the identity given by:\[\frac{1-\cos \alpha}{\sin \alpha}=\frac{\sin \alpha}{1+\cos \alpha}\]To verify this, we must show that both sides simplify to the same expression or value.

Step 2: Simplify the Left Side

We begin by simplifying the left side of the equation:\[\frac{1-\cos \alpha}{\sin \alpha}\]Rewrite it as:\[\frac{1-\cos^2 \alpha}{\sin \alpha (1+\cos \alpha)}\]Since \(1-\cos^2 \alpha = \sin^2 \alpha\), this becomes:\[\frac{\sin^2 \alpha}{\sin \alpha (1+\cos \alpha)} = \frac{\sin \alpha}{1+\cos \alpha}\]The left side simplifies to the same form as the right side.

Step 3: Simplify the Right Side

For completeness, simplify the right side of the equation:\[\frac{\sin \alpha}{1+\cos \alpha}\]Since no further simplification is necessary, it is clear that both sides of the equation are identical after simplifying the left side.

Step 4: Conclude the Verification

Since the simplified forms of both the left and right sides of the equation are identical:\[\frac{\sin \alpha}{1+\cos \alpha}\]This confirms that the original identity holds true, and the identity is verified.

Key Concepts

Verification of Trigonometric Identities

Verifying trigonometric identities involves demonstrating that two different expressions are actually equal to each other. This is crucial in trigonometry as identities form the backbone of simplifying and solving various problems. To verify an identity, follow these steps:

• Identify the two sides of the identity you need to compare. On the left side, we started with \( \frac{1-\cos \alpha}{\sin \alpha} \).
• Determine if one side is more complex than the other and begin simplifying that side. For example, the left side was more complex in this case.
• Use known trigonometric identities, like the Pythagorean identity, to simplify the expression. We used \( 1-\cos^2 \alpha = \sin^2 \alpha \) to transform the expression.
• Determine if the simplified form of one expression matches the other side of the original equation.

By simplifying the left side to \( \frac{\sin \alpha}{1+\cos \alpha} \), we validated that both sides of the identity are indeed equivalent.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions is an essential skill to master, as it allows for easier handling of complex equations. The goal is to reduce a trigonometric expression to its simplest form using a combination of algebraic techniques and trigonometric identities.

• Factorizing expressions is crucial. Our original expression, \( \frac{1-\cos \alpha}{\sin \alpha} \), benefited from recognizing the difference of squares; \( 1-\cos^2 \alpha \) transformed into \( \sin^2 \alpha \).
• Cancel out common terms where possible. This reduces the complexity of the expression and makes calculations smoother, as seen when we reduced \( \frac{\sin^2 \alpha}{\sin \alpha} \) to \( \sin \alpha \).
• Always look for opportunities to use trigonometric identities to turn a long expression into something more manageable.

By the end of this process, the expression becomes much simpler, making it easier to work with, especially in verifying identities or solving equations.

Pythagorean Identities

The Pythagorean identities are foundational in trigonometry. They express essential relationships between sine and cosine and can transform and simplify complex expressions.The primary Pythagorean identity is:\[ \sin^2 \theta + \cos^2 \theta = 1\]From this core identity, we can derive two others by manipulating this expression:

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

These identities are particularly useful because they allow you to convert between sine and cosine, depending on what the problem requires. For instance, in our verification task, we used \( 1-\cos^2 \alpha = \sin^2 \alpha \) to transform the left side of the initial identity into a more comparable form.Understanding and applying these identities can simplify complex trigonometric problems and verify identities more effectively, making them indispensable tools for any math student.