Problem 35 - Algebra 2

Question

Simplify each trigonometric expression. $$ \sec \theta\left(1+\cot ^{2} \theta\right) $$

Step-by-Step Solution

Answer

The simplification of the given expression \(\sec \theta\left(1+\cot ^{2} \theta\right)\) is \(\frac{1 + \cot^2 \theta}{\cos \theta}\).

Step 1: Understanding Reciprocal Relationships

The trigonometric function \(\sec \theta\) is the reciprocal of \(\cos \theta\). That is, \(\sec \theta = \frac{1}{\cos \theta}\). Also, \(\cot \theta = \frac{\cos \theta}{\sin \theta}\), or it can be represented in terms of sine and cosine as \(\cot ^{2} \theta = \frac{\cos ^{2} \theta}{\sin ^2 \theta}\). These expressions will be used to simplify the expression.

Step 2: Simplification Using Trig Identity

The identity \(\sin ^{2} \theta + \cos ^{2} \theta = 1\) is a fundamental identity in trigonometry. By subtracting \(\cos ^{2} \theta\) from both sides, we get \(\sin ^{2} \theta = 1 - \cos ^{2} \theta\). This important relation would simplify the denominator of \(\cot ^{2} \theta\) by replacing \(\sin ^{2} \theta\) with \(1 - \cos ^{2} \theta\).

Step 3: Substitute and Simplify

Substituting the identities from Steps 1 and 2 into the expression: \(\sec \theta(1 + \cot^{2} \theta) = \sec \theta + \sec \theta \cdot \cot^{2} \theta\). Substituting the results from step 1: \( = \frac{1}{\cos \theta} + \frac{1}{\cos \theta} \cdot \frac{\cos ^{2} \theta}{1 - \cos ^{2} \theta}\). Simplifying the above expression gives us: \( = \frac{1}{\cos \theta} + \frac{\cos \theta}{1 - \cos ^{2} \theta}\). The second term can then be simplified using the identity \(1 - \cos ^2 \theta = \sin ^2 \theta\): \( = \frac{1}{\cos \theta} + \frac{\cos \theta}{\sin ^2 \theta}\). This simplifies to: \( = \frac{1 + \cot^2 \theta}{\cos \theta}\).

Key Concepts

Reciprocal Relationships

In trigonometry, understanding reciprocal relationships is essential for simplifying expressions. These relationships describe how one function can be expressed as the inverse, or reciprocal, of another. For instance, the secant function, denoted as \( \sec \theta \), is the reciprocal of the cosine function. In mathematical terms, this means:

• \( \sec \theta = \frac{1}{\cos \theta} \)

Similarly, the cotangent function \( \cot \theta \) relates to sine and cosine as:

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

Using these reciprocal identities allows us to express trigonometric functions in different forms, which can make them easier to manipulate and simplify. For example, understanding these relationships helped us to identify \( \cot^2 \theta \) in the original expression as \( \frac{\cos^2 \theta}{\sin^2 \theta} \). Knowing these connections is vital when working with trigonometric identities and simplifying complex expressions.

Pythagorean Identity

The Pythagorean Identity is one of the cornerstone principles in trigonometry, representing the relation between sine and cosine.

• The core identity is: \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity can be manipulated to express \( \sin^2 \theta \) in terms of \( \cos^2 \theta \) and vice versa. Through simple algebra, you can subtract \( \cos^2 \theta \) from both sides of the identity to solve for \( \sin^2 \theta \):

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

This transformation is incredibly useful when we're trying to simplify trigonometric expressions involving squares. For example, in the original solution, this identity allowed us to replace \( \sin^2 \theta \) with \( 1 - \cos^2 \theta \), helping reduce the complexity of the expression. Understanding and applying the Pythagorean Identity is crucial when attempting to simplify or verify other trigonometric identities.

Trigonometric Simplification

Trigonometric simplification involves using known identities and relationships to reduce complex expressions into more manageable forms. It often requires creativity in substituting equivalents and rearranging terms to expose simpler structures. The original exercise provides a clear illustration of this process.First, we make use of reciprocal relationships such as expressing \( \sec \theta \) as \( \frac{1}{\cos \theta} \). This substitution helps line up terms for potential simplification. Similarly, we identify \( \cot^2 \theta \) as \( \frac{\cos^2 \theta}{\sin^2 \theta} \).Next, employing the Pythagorean Identity \( \sin^2 \theta = 1 - \cos^2 \theta \), we can ensure that the denominator in \( \cot^2 \theta \) aligns with an identity that simplifies further. This allows the expression \( \sec \theta(1 + \cot^2 \theta) \) to break down into more subtle terms that ultimately lead us to a simpler form \( \frac{1 + \cot^2 \theta}{\cos \theta} \).

• Utilize identities wisely to eliminate common terms.
• Look for ways to rearrange terms to showcase simple patterns.

The art of trigonometric simplification doesn't just rest on knowing identities but also knowing when and how to apply them efficiently.