Problem 48 - 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.) $$\cot ^{2} \theta\left(1+\tan ^{2} \theta\right)$$

Step-by-Step Solution

Answer

The simplified expression is \( \csc^2 \theta \).

Step 1: Express Logarithms in Trigonometric Functions

The given expression is \( \cot^2 \theta (1 + \tan^2 \theta) \). To express it in terms of sine and cosine, recall the definitions of cotangent and tangent.\( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Step 2: Substitute Definitions into Expression

Substitute the definitions of cotangent and tangent into the expression:\[ \cot^2 \theta = \left(\frac{\cos \theta}{\sin \theta}\right)^2 = \frac{\cos^2 \theta}{\sin^2 \theta} \]\[ 1 + \tan^2 \theta = 1 + \left(\frac{\sin \theta}{\cos \theta}\right)^2 = 1 + \frac{\sin^2 \theta}{\cos^2 \theta} \].

Step 3: Simplify the Trigonometric Expression

The expression becomes:\[ \frac{\cos^2 \theta}{\sin^2 \theta} \left(1 + \frac{\sin^2 \theta}{\cos^2 \theta}\right) \]Now simplify the term inside the parentheses:\[ 1 + \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{\cos^2 \theta}{\cos^2 \theta} + \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta}\].

Step 4: Use Pythagorean Identity

Apply the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \) to simplify:\[ \frac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta} = \sec^2 \theta \].

Step 5: Full Simplification of Expression

Therefore, the expression becomes:\[ \frac{\cos^2 \theta}{\sin^2 \theta} \cdot \sec^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \cdot \frac{1}{\cos^2 \theta} \]This simplifies to \( \frac{1}{\sin^2 \theta} = \csc^2 \theta \).

Key Concepts

Understanding Cotangent in Trigonometry

In trigonometry, the cotangent function is essential for expressing angles in terms of sine and cosine. It is the reciprocal of the tangent function. Specifically, for any angle \( \theta \), cotangent is represented as:

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

By understanding cotangent in this way, you can rewrite expressions to focus on sine and cosine, which are the foundational trigonometric functions. This approach simplifies complex expressions and helps solve trigonometric exercises efficiently.
Consistent with these simplifications, you often encounter expressions involving \( \cot^2 \theta = \left(\frac{\cos \theta}{\sin \theta}\right)^2 = \frac{\cos^2 \theta}{\sin^2 \theta} \), which allow conversions between different trigonometric identities.

Tangent and Its Role in Simplifying Expressions

The tangent function is another critical trigonometric function and is defined as the ratio of sine to cosine for an angle \( \theta \). It is expressed as:

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

In exercises, such as the one we're considering, expressing equations in terms of tangent can lead to elegant simplifications. When we have \( 1 + \tan^2 \theta \), this can be rephrased into:

• \( 1 + \left(\frac{\sin \theta}{\cos \theta}\right)^2 = 1 + \frac{\sin^2 \theta}{\cos^2 \theta} \)

This set-up often calls for the Pythagorean identity in the following steps. Understanding tangent and its square, \( \tan^2 \theta \), makes it easy to manipulate expressions by aligning them with this foundational trigonometric formula.

The Crucial Role of the Pythagorean Identity

The Pythagorean Identity is a foundation of trigonometry. It states that for any angle \( \theta \):

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

This identity is crucial in simplifying trigonometric expressions, such as the one in our example. By substituting the identity into complex equations, you reduce their complexity significantly, dramatically simplifying solutions. In our problem, applying this identity alters:

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

Using the Pythagorean Identity, the equation further simplifies to \( \sec^2 \theta \), enabling straightforward continuation to final results. Recognizing and using this identity transforms complicated trigonometric expressions into manageable pieces. This is why it lies at the heart of many trigonometric simplifications.