Problem 106 - A Graphical Approach to Precalculus with Limits

Question

Verify that equation is an identity. \(\frac{\sin \theta}{1-\cos \theta}-\frac{\sin \theta \cos \theta}{1+\cos \theta}=\csc \theta\left(1+\cos ^{2} \theta\right)\)

Step-by-Step Solution

Answer

The given equation is an identity.

Step 1: Convert to a Common Denominator

The left-hand side of the equation is \( \frac{\sin \theta}{1-\cos \theta} - \frac{\sin \theta \cos \theta}{1+\cos \theta} \). To combine these fractions, find a common denominator, which is \((1-\cos \theta)(1+\cos \theta)\). The expression becomes \( \frac{\sin\theta(1+\cos\theta) - \sin\theta\cos\theta(1-\cos\theta)}{(1-\cos\theta)(1+\cos\theta)} \).

Step 2: Simplify the Numerators

Distribute and simplify the numerators. You have: \[ \sin \theta (1 + \cos \theta) = \sin \theta + \sin \theta \cos \theta \] \[ \sin \theta \cos \theta (1 - \cos \theta) = \sin \theta \cos \theta - \sin \theta \cos^2 \theta \] Subtracting these gives: \( \sin \theta + \sin \theta \cos \theta - \sin \theta \cos \theta + \sin \theta \cos^2 \theta \). Simplifying this results in \( \sin \theta (1 + \cos^2 \theta) \).

Step 3: Simplify the Denominator

Rewrite the denominator using the identity \((1-\cos\theta)(1+\cos\theta) = 1-\cos^2\theta\), which is equal to \(\sin^2\theta\). Thus, the left-hand side simplifies to \( \frac{\sin\theta(1+\cos^2\theta)}{\sin^2\theta} \).

Step 4: Simplify the Fraction

Cancel \( \sin \theta \) in the numerator and \( \sin^2 \theta \) in the denominator, resulting in \( \csc \theta (1 + \cos^2 \theta) \).

Step 5: Verify the Identity

The simplified left-hand side \( \csc \theta (1 + \cos^2 \theta) \) matches the right-hand side of the original equation, confirming it is an identity.

Key Concepts

Common Denominator

When working with fractions, finding a common denominator is crucial for combining them into a single expression. In trigonometry, as with numerical fractions, the common denominator ensures the subtraction or addition process is valid.
For the given identity \[\frac{\sin \theta}{1-\cos \theta}-\frac{\sin \theta \cos \theta}{1+\cos \theta}=\csc \theta\left(1+\cos ^{2} \theta\right),\] the separate fractions have different denominators:

• \(1-\cos \theta\), and
• \(1+\cos \theta\).

When combining these fractions, we need the product of these denominators: \((1-\cos \theta)(1+\cos \theta)\). This allows us to express each fraction with this shared denominator, providing a basis for further manipulation.

Simplification

Simplification is a process of reducing expressions to their most basic form. This process often involves combining like terms and applying algebraic identities to eliminate complex terms.
In our equation, once the fractions have a common denominator, the numerators \[\sin \theta (1 + \cos \theta) \quad \text{and} \quad \sin \theta \cos \theta (1 - \cos \theta)\] need to be expanded and simplified.
Through distribution, terms like \(\sin \theta + \sin \theta \cos \theta - \sin \theta \cos \theta + \sin \theta \cos^2 \theta\) appear. By recognizing and cancelling out terms, the expression simplifies to \(\sin \theta (1 + \cos^2 \theta)\).
Effective simplification reveals the core structure of the expression, enabling easier manipulation and understanding.

Trigonometric Functions

Trigonometric functions like sine, cosine, and cosecant are foundational elements in trigonometry.
In the identity from the exercise, \(\sin \theta\) and \(\cos \theta\) dominate the expressions on the left, while \(\csc \theta\) appears on the right.
Cosecant, \(\csc \theta\), is the reciprocal of sine: \(\csc \theta = \frac{1}{\sin \theta}\).
Understanding how these functions relate helps you manipulate and simplify expressions more effectively. They enable the transformation from complex fraction forms into clear, verifiable identities. Trigonometric functions bridge the gap between diverse mathematical expressions due to their inherent properties and interrelationships.

Pythagorean Identity

The Pythagorean Identity is an essential and widely used relationship in trigonometry. It states \(\sin^2 \theta + \cos^2 \theta = 1\).
This identity is particularly useful when rewriting or simplifying expressions involving squares of sine and cosine.
In the provided solution, we use the Pythagorean Identity to revamp the denominator \((1 - \cos^2 \theta)\), which simplifies to \(\sin^2 \theta\).
This transformation highlights how seemingly complex expressions can be made simpler and more tractable. The Pythagorean Identity not only aids in simplification but also in identifying equivalent expressions, confirming that complex-looking identities are indeed true.

• This identity is a foundation in verifying and transforming trigonometric expressions.