Problem 70 - Calculus for Biology and Medicine

Question

Show that the identity $$ 1+\cot ^{2} \theta=\csc ^{2} \theta $$ follows from $$ \sin ^{2} \theta+\cos ^{2} \theta=1 $$

Step-by-Step Solution

Answer

Using the Pythagorean identity, we derived \(1 + \cot^2 \theta = \csc^2 \theta\).

Step 1: Understanding the Pythagorean Identity

We start with the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity represents the fundamental relationship between sine and cosine for any angle \( \theta \).

Step 2: Express Cotangent and Cosecant in Terms of Sine and Cosine

Recall the definitions of cotangent and cosecant: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \). We aim to express \(1 + \cot^2 \theta\) in terms of sine and cosine.

Step 3: Substitute Cotangent Square

Substitute \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) into the equation \( \cot^2 \theta = \left(\frac{\cos \theta}{\sin \theta}\right)^2 = \frac{\cos^2 \theta}{\sin^2 \theta} \).

Step 4: Rewrite 1 + Cotangent Square

Start with \( 1 + \cot^2 \theta = 1 + \frac{\cos^2 \theta}{\sin^2 \theta} \). Recognize that \( 1 = \frac{\sin^2 \theta}{\sin^2 \theta} \). Thus, \( 1 + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta} \).

Step 5: Utilize the Pythagorean Identity Again

Substitute \( \sin^2 \theta + \cos^2 \theta = 1 \) into \( \frac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta} \) to get \( \frac{1}{\sin^2 \theta} \).

Step 6: Relate to Cosecant Square

Realize that \( \frac{1}{\sin^2 \theta} = \csc^2 \theta \) by definition. Therefore, \( 1 + \cot^2 \theta = \csc^2 \theta \), completing our proof.

Key Concepts

Pythagorean Identity

The Pythagorean identity is one of the foundational identities in trigonometry. It's expressed as: \[ \sin^2 \theta + \cos^2 \theta = 1 \]This means that for any angle \( \theta \), the sum of the square of sine and the square of cosine is always equal to 1. This identity is analogous to the Pythagorean theorem for right triangles, hence the name. When we use the Pythagorean identity in trigonometric proofs, it helps us relate different trigonometric functions, making it simpler to solve problems. Additionally, this identity forms the basis for deriving other important identities in trigonometry. Break this total into parts by using known trigonometric functions: - The sine function, \( \sin \theta \), represents the opposite side over the hypotenuse in a right triangle.- The cosine function, \( \cos \theta \), represents the adjacent side over the hypotenuse. Understanding these relationships is key to transforming and rearranging trigonometric equations.

Cotangent and Cosecant

In trigonometry, cotangent and cosecant are reciprocal identities derived from the basic sine and cosine functions. These reciprocal functions aid in simplifying computations involving triangles and angles.- Cotangent is defined as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), meaning it is the ratio of the adjacent side to the opposite side in a right triangle, and can be seen as the inverse of the tangent function.- Cosecant is defined as \( \csc \theta = \frac{1}{\sin \theta} \), which is the reciprocal of the sine function.These definitions are particularly useful when manipulating expressions and proving identities. Understanding these functions allows us to express one trigonometric function in terms of another, facilitating the simplification of complex expressions. For example, in the identity \( 1 + \cot^2 \theta = \csc^2 \theta \), we start by expressing \( \cot^2 \theta \) and \( \csc^2 \theta \) in terms of sine and cosine to verify and simplify the identity. The expressibility of these functions in terms of sine and cosine underlines their inherent interconnectedness within trigonometry.

Trigonometric Proofs

Trigonometric proofs are exercises that involve validating trigonometric identities through logical reasoning and transformation of equations. The goal of a trigonometric proof is to show that two seemingly different expressions are, in fact, equivalent. Approaching a trigonometric proof involves:- **Understanding the Given Identity:** Recognize the identity you are trying to prove. For our exercise, it's showing that \( 1 + \cot^2 \theta = \csc^2 \theta \).- **Express in Basic Terms:** Break down complex trigonometric functions into their sine and cosine components. This helps in visualizing connections and transformations.- **Use Known Identities:** Utilize well-established identities, such as the Pythagorean identity, to simplify terms.- **Step-by-Step Manipulation:** Conduct transformations step-by-step, maintaining equality at each stage, which leads to the final expression. In our example, we began with the Pythagorean identity. We expressed \( \cot^2 \theta \) and simplified using known relationships to conclude with \( \csc^2 \theta \). This methodical approach is crucial for success in proving trigonometric identities and deepening our understanding of the intricate web of trigonometric equations.