Problem 1 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\csc \theta-\sin \theta=\cot \theta \cos \theta$$

Step-by-Step Solution

Answer

The identity \(\csc \theta - \sin \theta = \cot \theta \cos \theta\) is verified as both sides simplify to \(\frac{\cos^2 \theta}{\sin \theta}\).

Step 1: Rewrite Trigonometric Functions

Rewrite the cosecant and cotangent in terms of sine and cosine: \[ \csc \theta = \frac{1}{\sin \theta} \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta} \]

Step 2: Substitute into the Identity

Substitute the rewritten trigonometric functions into the original identity: \[ \frac{1}{\sin \theta} - \sin \theta = \frac{\cos \theta}{\sin \theta} \cdot \cos \theta \]

Step 3: Simplify the Left Side

Combine the terms on the left side over a common denominator: \[ \frac{1}{\sin \theta} - \sin \theta = \frac{1 - \sin^2 \theta}{\sin \theta} \]

Step 4: Use Pythagorean Identity

Recognize that \(1 - \sin^2 \theta = \cos^2 \theta\) using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute this into the equation:\[ \frac{\cos^2 \theta}{\sin \theta} \]

Step 5: Simplify the Right Side

Multiply the terms on the right side:\[ \frac{\cos \theta \cdot \cos \theta}{\sin \theta} = \frac{\cos^2 \theta}{\sin \theta} \]

Step 6: Compare Both Sides

Observe that both sides simplify to \(\frac{\cos^2 \theta}{\sin \theta}\), thereby verifying the identity. This confirms that:\[ \frac{1 - \sin^2 \theta}{\sin \theta} = \frac{\cos^2 \theta}{\sin \theta}\]

Key Concepts

Cosecant

The term 'Cosecant' might sound complex, but it simply represents the reciprocal of the sine function in trigonometry. In mathematical terms, the cosecant of an angle \( \theta \) is defined as:

• \( \csc \theta = \frac{1}{\sin \theta} \)

This relationship is crucial when solving trigonometric identities because it allows us to express the cosecant in terms of the sine, making equations easier to analyze and manipulate.
For example, if we have \( \csc \theta \) in an equation, we can substitute it with \( \frac{1}{\sin \theta} \) to potentially simplify expressions and solve identities.
Using reciprocal relationships like that of cosecant not only aids in verifying identities but is also useful in solving real-world problems involving angles and lengths.

Cotangent

Cotangent is another trigonometric function that is often encountered in equations and identities. It is the reciprocal of the tangent function, defined as:

• \( \cot \theta = \frac{1}{\tan \theta} \), which can also be written as \( \frac{\cos \theta}{\sin \theta} \)

Understanding how to express cotangent in terms of sine and cosine—specifically \( \frac{\cos \theta}{\sin \theta} \)—is instrumental in solving and simplifying trigonometric equations.
In the context of verifying identities, recognizing alternate forms of cotangent helps to transform the given terms into forms that are easier to compare or combine with other terms in the equation.
This expression is particularly helpful when you need to rewrite equations by using common denominators or when manipulating numerators and denominators to simplify trigonometric expressions.

Pythagorean Identity

The Pythagorean Identity is a fundamental identity in trigonometry that relates sine and cosine functions. The most common form of this identity is:

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

This identity can be rearranged to find expressions for \( 1 - \sin^2 \theta \) or \( 1 - \cos^2 \theta \). As seen in our step-by-step solution, we take advantage of this property:

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

Understanding and applying the Pythagorean Identity is central to verifying trigonometric identities because it provides a method to rewrite parts of an equation in terms of other functions.
This process is invaluable when combining or simplifying expressions and is widely used in calculus and other advanced mathematics to prove other identities or solve equations involving trigonometric functions. This identity ensures we can switch between \( \sin^2 \theta \) and \( \cos^2 \theta \) knowing they together form a complete entity of 1.