Problem 73 - Precalculus with Limits

Question

In Exercises 71 - 74, perform the multiplication and use the fundamental identities to simplify. There is more than one correct form of each answer. \( \left(2 \csc x + 2\right)\left(2 \csc x - 2\right) \)

Step-by-Step Solution

Answer

After simplification, the expression \( \left(2 \csc x + 2\right)\left(2 \csc x - 2\right) \) equals \(4 \cot^2 x\).

Step 1: Apply Difference of Squares

The first step is to multiply the two expressions by applying the formula for the difference of squares. So, \((2 \csc x + 2)(2 \csc x - 2)\) becomes \((2 \csc x)^2 - (2)^2\), which simplifies to \(4 \csc^2 x - 4\).

Step 2: Apply Fundamental Identity

Next, we substitute the fundamental identity \(\csc^2 x = 1+ \cot^2 x\) into the expression. Thus, \(4 \csc^2 x - 4\) becomes \(4(1 + \cot^2 x) - 4\).

Step 3: Simplify Expression

Finally, we simplify the expression to its simplest form by distributing and combining like terms. Distributing the \(4\) gives \(4+ 4 \cot^2 x - 4\), and subtracting the \(4\)s results in \(4 \cot^2 x\).

Key Concepts

Difference of Squares

The difference of squares is a useful algebraic identity that allows you to quickly simplify expressions where one term is subtracted from another. It follows the formula:

• \[(a - b)(a + b) = a^2 - b^2\]

This formula is derived from the distributive property. It signifies that the product of a sum and a difference of two terms is equal to the difference of their squares. In our exercise, we applied this identity to the expression

• \[(2 \csc x + 2)(2 \csc x - 2)\]

Using the difference of squares formula, it simplified to

• \[(2 \csc x)^2 - 2^2\]

which further simplifies to

• \[4 \csc^2 x - 4\]

Mastering this identity saves time and effort when simplifying polynomials and is essential in various mathematical problems.

Cosecant Function

The cosecant function, often written as \(\csc x\), is the reciprocal of the sine function. It is given by:

• \[\csc x = \frac{1}{\sin x}\]

Being the reciprocal, the cosecant function is undefined wherever the sine is zero. This happens at multiples of \(\pi\) (180 degrees), where the graph approaches vertical asymptotes.

The cosecant function plays a crucial role in trigonometry, much like the other fundamental trigonometric functions, and is particularly important in the example given.By understanding its reciprocal relationship with sine, it becomes easier to manage transformations and computations that involve \(\csc x\).

A key identity involving the cosecant function is:

• \[\csc^2 x = 1 + \cot^2 x\]

This identity helps in simplifying complex trigonometric expressions, as seen in our solution step that replaces \(\csc^2 x\) with \(1 + \cot^2 x\).

Cotangent Function

The cotangent function, expressed as \(\cot x\), is another reciprocal trigonometric function. It is defined as:

• \[\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}\]

Like the cosecant function, cotangent is undefined at points where its reciprocal, the tangent function, equals zero. Specifically, this occurs at the angles \(\frac{\pi}{2}, \frac{3\pi}{2},\) and so on.

In trigonometry, the

• \(1 + \cot^2 x = \csc^2 x\)

identity proves helpful, as it allows for transformations between different trigonometric functions.

During simplifications, incorporating the identity helps manage expressions involving cotangent more efficiently.

Understanding how cotangent relates to the other functions and identities provides a comprehensive view, making it easier to tackle varied trigonometric problems.