Problem 61 - A Graphical Approach to Precalculus with Limits

Question

Perform indicated operation and simplify the result. $$\cot \theta+\frac{1}{\cot \theta}$$

Step-by-Step Solution

Answer

The simplified result is \( \frac{2}{\sin(2\theta)} \).

Step 1: Identify the Trigonometric Identity

Recall the trigonometric identity for cotangent in terms of sine and cosine. The cotangent of an angle \( \theta \) is given by \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Similarly, the reciprocal of cotangent is \( \frac{1}{\cot \theta} = \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Step 2: Express Each Term in Sin and Cos

Using the identity \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and \( \frac{1}{\cot \theta} = \frac{\sin \theta}{\cos \theta} \), express the original expression as:\[ \cot \theta + \frac{1}{\cot \theta} = \frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta} \].

Step 3: Find a Common Denominator

In order to add the two fractions, we need to find a common denominator. The common denominator for \( \frac{\cos \theta}{\sin \theta} \) and \( \frac{\sin \theta}{\cos \theta} \) is \( \sin \theta \cos \theta \).

Step 4: Rewrite Fractions with Common Denominator

Rewriting each fraction with the common denominator:\[\frac{\cos \theta \cdot \cos \theta}{\sin \theta \cos \theta} + \frac{\sin \theta \cdot \sin \theta}{\sin \theta \cos \theta}\]=\[\frac{\cos^2 \theta}{\sin \theta \cos \theta} + \frac{\sin^2 \theta}{\sin \theta \cos \theta}\].

Step 5: Combine the Fractions

Now, combine the two fractions:\[\frac{\cos^2 \theta + \sin^2 \theta}{\sin \theta \cos \theta}\].

Step 6: Simplify Using Pythagorean Identity

Recall that one of the Pythagorean identities is \( \cos^2 \theta + \sin^2 \theta = 1 \). Applying this identity to the numerator gives:\[\frac{1}{\sin \theta \cos \theta}\].

Step 7: Simplify Further

Recognize that \( \sin \theta \cos \theta \) in the denominator can be remembered as \( \frac{1}{2} \sin(2\theta) \) in trigonometric identities. Therefore:\[ \frac{1}{\sin \theta \cos \theta} = \frac{2}{\sin(2\theta)} \].

Key Concepts

Cotangent

Understanding the concept of cotangent is key when dealing with trigonometric expressions. The cotangent of an angle, denoted as \( \cot \theta \), is defined as the ratio of the adjacent side to the opposite side in a right triangle, based on the angle \( \theta \). Mathematically, it is expressed in terms of sine and cosine as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). This relationship helps us transition between different trigonometric functions and is especially useful in verifying identities and simplifying expressions.
Knowing these relationships aids in transforming complex trigonometric problems into simpler forms, such as expressing terms in terms of sine and cosine, which sets the stage for applying other identities effectively.

Pythagorean Identity

The Pythagorean Identity is one of the cornerstone relationships in trigonometry. It states that for any angle \( \theta \), the sum of the square of sine and cosine is always equal to one: \( \cos^2 \theta + \sin^2 \theta = 1 \).
This identity is derived from the Pythagorean theorem applied in the context of the unit circle. These identities are extremely helpful in simplifying trigonometric expressions, as they allow substitution and reduction of terms that might otherwise complicate the equation.
The identity was crucial in the original problem solution, where the sum \( \cos^2 \theta + \sin^2 \theta \) was simplified to 1, thus making further simplification of the expression much easier.

Common Denominator

When adding or subtracting fractions, finding a common denominator is an essential step. In trigonometry, this often involves expressing fractions in terms of sine and cosine, and then finding a common base for addition or subtraction.
For the expression \( \frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta} \), the denominators \( \sin \theta \) and \( \cos \theta \) necessitate multiplying each term by the other fraction's denominator to create a common denominator of \( \sin \theta \cos \theta \).
With a common denominator, terms can be directly combined, simplifying complex expressions and facilitating the application of identities. Practicing this technique ensures you can tackle similar problems with confidence.

Reciprocal Identities

Reciprocal identities are fundamental relationships in trigonometry that link angles with their reciprocal functions. For instance, the reciprocal of cotangent, \( \cot \theta \), is \( \tan \theta \), expressed as \( \frac{1}{\cot \theta} = \tan \theta = \frac{\sin \theta}{\cos \theta} \).
These identities allow the transformation of an expression involving one trigonometric function into another, potentially more manageable form. In the original exercise, understanding that the reciprocal of \( \cot \theta \) is \( \tan \theta \) as \( \frac{\sin \theta}{\cos \theta} \) was crucial for eventually expressing terms in forms that facilitate the application of further identities.
This interconnected system of identities enriches the toolkit for solving trigonometric problems efficiently and effectively.