Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$\frac{\cos ^{2} \theta-\sin ^{2} \theta}{\sin \theta \cos \theta}$$

Step-by-Step Solution

Answer

The simplified expression is \(2\cot(2\theta)\).

Step 1: Recognize Trigonometric Identity

Start by recognizing the numerator \( \cos^2 \theta - \sin^2 \theta \). This is a known trigonometric identity for \( \cos(2\theta) \). Thus, rewrite it as \( \cos(2\theta) \).

Step 2: Simplify the Expression

The given expression can now be written as \( \frac{\cos(2\theta)}{\sin\theta \cos\theta} \).

Step 3: Use Double Angle Identity for Sine

Recall that \( \sin(2\theta) = 2\sin\theta\cos\theta \). Substitute \( 2\sin\theta\cos\theta \) for the denominator. The expression becomes \( \frac{\cos(2\theta)}{\frac{1}{2}\sin(2\theta)} \).

Step 4: Simplify the Fraction

Since the denominator is \( \frac{1}{2}\sin(2\theta) \), the fraction can be rewritten as \( 2\frac{\cos(2\theta)}{\sin(2\theta)} \).

Step 5: Write in Terms of Cotangent

Recognize that \( \frac{\cos(2\theta)}{\sin(2\theta)} \) is the definition of \( \cot(2\theta) \). Thus, the expression simplifies to \( 2\cot(2\theta) \).

Key Concepts

Double Angle Formulas

Double angle formulas are a set of important identities in trigonometry. They help us express trigonometric functions of angles like \(2\theta\) in terms of the function at \(\theta\). Specifically, they not only simplify complex expressions but also have practical applications in solving equations and transformations.
One such double angle formula is for cosine:

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

This formula transforms a difference of squares in sine and cosine into a simple cosine of a doubled angle. Also, we often utilize other versions of this formula, including:

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

These variations allow greater flexibility based on the problem context. By interpreting the numerator \( \cos^2\theta - \sin^2\theta \) using the double angle formula, we can simplify expressions significantly. In the exercise above, this simplification transforms the original complex expression into a manageable form involving \(\cos(2\theta)\).

Sine and Cosine Functions

The sine and cosine functions are fundamental components of trigonometry. They describe the relationship between angles in a right triangle and the ratios of their corresponding sides. These functions oscillate between -1 and 1 and are periodic in nature.
In the context of this exercise:

• \(\sin(\theta)\) represents the ratio of the opposite side to the hypotenuse in a right triangle.
• \(\cos(\theta)\) represents the ratio of the adjacent side to the hypotenuse in a right triangle.

Understanding these functions is crucial for simplifying trigonometric expressions. Notably, they form the building blocks for many identities, including the double angle formulas.
In the given expression, sine and cosine functions provided a pathway through simplification. The denominator, \(\sin\theta \cos\theta\), was rewritten using the double angle formula for sine, \(\sin(2\theta) = 2\sin\theta \cos\theta\). This allowed the expression to be simplified into a more straightforward cosine and sine relationship.

Cotangent Function

The cotangent function, denoted as \(\cot\), is a trigonometric function that is lesser-known than sine and cosine but equally important in certain contexts. The cotangent of an angle \(\theta\) is the reciprocal of the tangent function:

• \(\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}\)

In this article, understanding \(\cot(2\theta)\) is crucial to simplifying the initial complex expression. By recognizing that the fraction \(\frac{\cos(2\theta)}{\sin(2\theta)}\) corresponds to \(\cot(2\theta)\), we can further compress the expression.
As demonstrated in the exercise, the final simplified expression was derived as \(2 \cot(2\theta)\). This showcases the utility of the cotangent function in translating complex trigonometric forms into more elegant and simplified solutions.
Recognizing and effectively using such trigonometric identities is a valuable skill. It aids in solving equations where direct computation could be daunting.