Problem 36 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\csc x[\csc x+\sin (-x)]=\cot ^{2} x$$

Step-by-Step Solution

Answer

The identity is verified by simplifying the left-hand side to match the right-hand side, \(\cot^2 x\).

Step 1: Understand the terms

First, recognize the trigonometric functions in the expression. We have \(\csc x\) (cosecant), \(\sin(-x)\) (sine of negative angle, which is \(-\sin x\) since sine is an odd function), and \(\cot x\) (cotangent).

Step 2: Simplify the original expression

Start simplifying the left-hand side (LHS) of the equation: \[ \csc x [\csc x + \sin(-x)] = \csc x [\csc x - \sin x] \]This simplifies to:\[ \csc^2 x - \csc x \sin x \]

Step 3: Substitute identities

Replace \(\csc x\) with \(1/\sin x\):\[ \csc^2 x - \csc x \sin x = \frac{1}{\sin^2 x} - \frac{1}{\sin x} \sin x \]which simplifies to:\[ \frac{1}{\sin^2 x} - 1 \]

Step 4: Express LHS in terms of cotangent

Realize that \(1 - \sin^2 x = \cos^2 x\):\[ \frac{1}{\sin^2 x} - 1 = \frac{1 - \sin^2 x}{\sin^2 x} = \frac{\cos^2 x}{\sin^2 x} \]

Step 5: Final simplification and compare sides

Recognize \(\frac{\cos^2 x}{\sin^2 x}\) as \(\cot^2 x\): \[ \frac{\cos^2 x}{\sin^2 x} = \cot^2 x \] Thus, \(\csc x [\csc x + \sin(-x)] = \cot^2 x\). The identity is verified.

Key Concepts

Cosecant

The cosecant function, denoted as \( \csc x \), is one of the lesser-known trigonometric functions but is crucial for solving complex trigonometric problems. Unlike the sine function, which is more common, the cosecant is the reciprocal. Specifically, it is the reciprocal of the sine function. This means that \( \csc x = \frac{1}{\sin x} \).

Because of this reciprocal relationship, understanding the properties and behaviors of the sine function will inherently give insights into how the cosecant works. For instance, where sine is zero, cosecant is undefined, leading to interesting behavior of the graph.

In problems involving trigonometric identities, the ability to switch between \( \csc x \) and \( \sin x \) can provide various pathways to simplify or transform expressions in helpful ways.

Cotangent

Cotangent, represented by \( \cot x \), is another function often encountered in trigonometric identities. Like the cosecant is to sine, cotangent is similarly the reciprocal of tangent: \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \).

This relationship directly connects cotangent with the sine and cosine functions, allowing it to be expressed in terms of these fundamental functions easily. Understanding that cotangent is a ratio of the cosine and sine function values helps visualize and manipulate expressions during trigonometric simplifications.

When working through identities like the one given in the exercise, recognizing and substituting \( \cot^2 x \) can simplify expressions significantly, especially as it relates to \( \frac{\cos^2 x}{\sin^2 x} \). This highlights the utility in deeply understanding function relationships in trigonometry.

Sine Function

The sine function, often written as \( \sin x \), is arguably the most recognized trigonometric function, forming the basis of many trigonometric identities and transformations. Its role is central because it directly relates to the unit circle, where it represents the y-coordinate of a point at a given angle from the positive x-axis.

An interesting property of the sine function is its odd nature, meaning \( \sin(-x) = -\sin x \). This indicates that sine is symmetric about the origin in a graph of the function. This characteristic becomes useful when simplifying expressions involving negative angles, a common occurrence in identity verifications.

In trigonometric identities, expressing or isolating terms like \( \sin x \) can refocus a problem to its basic components, aiding in understanding and solving complex problems.

Trigonometric Simplification

Trigonometric simplification is a powerful process in solving and verifying trigonometric identities. It involves reducing expressions to a simpler form while retaining their equivalence. This often requires applying known identities and reciprocals.

A key strategy in simplification is recognizing interrelationships, such as how \( 1 - \sin^2 x = \cos^2 x \), derived from the Pythagorean identity. This is central when converting entire expressions into simpler forms or expressing them purely in terms of a single trigonometric function like cosine or sine.

In the process of simplification, converting fractions or removing terms by factoring common elements can reveal deeper structure within the expression, aiding in the resolution and understanding of complex trigonometric problems.