Problem 53 - Precalculus: Functions and Graphs

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\frac{\csc \theta}{\sec \theta}=\cot \theta$$

Step-by-Step Solution

Answer

The identity is verified: \(\frac{\csc\theta}{\sec\theta} = \cot\theta\).

Step 1: Recall Trigonometric Definitions

Recall the definitions of cosecant and secant. We have \( \csc \theta = \frac{1}{\sin \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \).

Step 2: Rewrite the Left-Hand Side

Substitute the trigonometric identities into the left-hand side \( \frac{\csc \theta}{\sec \theta} \) to get \( \frac{1}{\sin \theta} \div \frac{1}{\cos \theta} \).

Step 3: Simplify the Division

Simplify the expression \( \frac{1}{\sin \theta} \div \frac{1}{\cos \theta} \) as follows: multiply by the reciprocal to get \( \frac{1}{\sin \theta} \times \frac{\cos \theta}{1} = \frac{\cos \theta}{\sin \theta} \).

Step 4: Recognize the Result

Recognize that \( \frac{\cos \theta}{\sin \theta} \) is the definition of \( \cot \theta \). Therefore, the left-hand side \( \frac{\csc \theta}{\sec \theta} \) simplifies to \( \cot \theta \), which matches the right-hand side.

Key Concepts

Cosecant Function

The cosecant function, often abbreviated as \( \csc \theta \), is one of the lesser-known trigonometric functions but is crucial in various calculations. Its definition is precisely the reciprocal of the sine function, meaning:

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

This relationship implies that wherever sine is zero, the cosecant is undefined because division by zero is impossible.
Understanding \( \csc \theta \) can be very useful. It shows how the opposite side of a right triangle relates to the hypotenuse in a reciprocal manner.
Let's put it in a practical situation. If we have a triangle where \( \sin \theta = \frac{3}{5} \), then \( \csc \theta = \frac{5}{3} \). It provides a direct relationship to understanding reverse proportions within trigonometric contexts.

Secant Function

The secant function, denoted as \( \sec \theta \), is another trigonometric function based on reciprocity. It is the reciprocal of the cosine function:

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

This particular definition means that the secant value goes undefined when \( \cos \theta \) is zero.
An interesting aspect of the secant function is that it helps us assess the horizontal component's relation to the hypotenuse. When you think of the secant in practical terms, if \( \cos \theta = \frac{4}{5} \), then \( \sec \theta = \frac{5}{4} \).
This function, like the cosecant, is fundamental in understanding and solving identities and can frequently appear in trigonometric transformations and equations.

Cotangent Function

The cotangent function, abbreviated as \( \cot \theta \), is commonly used in various trigonometric identity verifications. It is defined as the reciprocal of the tangent function:

• \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \)

This means that \( \cot \theta \) gives you a measure of how the opposite side compares to the adjacent side in an inverse manner within a right triangle.
Understanding the cotangent function is essential for manipulating equations involving tangent and cotangent ratios. For example, if \( \tan \theta = 2 \), then \( \cot \theta = \frac{1}{2} \).
Using \( \cot \theta \) in identities can simplify problems and provide alternative approaches to solving trigonometric equations. Its role as a bridge between sine and cosine makes it integral for comparing and transforming trigonometric expressions.