Problem 54 - Precalculus with Limits

Question

In Exercises 53-60, (a) use a graphing utility to graph each side of the equation to determine whether the equation is an identity, (b) use the table feature of a graphing utility to determine whether the equation is an identity, and (c) confirm the results of parts (a) and (b) algebraically. \( \csc x (\csc x - \sin x) + \dfrac{\sin x - \cos x}{\sin x} + \cot x = \csc^2 x \)

Step-by-Step Solution

Answer

Yes, the given trigonometric equation is an identity.

Step 1: Rewrite the equation

Rewrite the equation separating each term to make the equation more clear. The given equation should look like this:\( \csc x \times \csc x - \csc x \times \sin x + \dfrac{\sin x - \cos x}{\sin x} + \cot x = \csc^2 x \)

Step 2: Simplify the equation

Now, simplify the equation replacing the identities:• Replace \( \csc x = \dfrac{1}{\sin x} \), so that, \( \csc x \times \csc x =(\dfrac{1}{\sin x})^2 \)• Replace \( \cot x = \dfrac{1}{\tan x} = \dfrac{\cos x}{\sin x} \)Rewrite the equation with the replacements:\((\dfrac{1}{\sin x})^2 - \dfrac{1}{1} + \dfrac{\sin x - \cos x}{\sin x} + \dfrac{\cos x}{\sin x} = (\dfrac{1}{\sin x})^2 \)

Step 3: Simplify further

Combine like terms to continue simplifying the equation:\((\dfrac{1}{\sin x})^2 + \dfrac{\sin x - \cos x + \cos x}{\sin x} - 1 = (\dfrac{1}{\sin x})^2 \)Simplify this further to:\((\dfrac{1}{\sin x})^2 + 1 - 1 = (\dfrac{1}{\sin x})^2 \)

Step 4: Checking for identity

After simplification, the left-hand side and the right-hand side of the equation are equal, confirming that the given equation is indeed an identity.

Key Concepts

Graphing Utility

Graphing utilities are powerful tools that allow us to visualize mathematical concepts and understanding the behavior of functions. In the realm of trigonometry, a graphing utility can be used to graph the separate components of a trigonometric identity, such as \( \text{csc} \text(x) \), \( \text{sin} \text(x) \), and \( \text{cot} \text(x) \). By graphing both sides of an equation, students can visually determine if the two sides coincide for all values of \( x \), which would imply the equation is an identity.

Most graphing calculators or software will have the ability to plot these functions over a specified interval. When using a graphing utility to confirm an identity, if the graphs of both sides of the equation are indistinguishable, it supports the hypothesis that the equation is a true identity. While not a formal proof, graphing provides a quick check and a strong visual understanding of the identity.

Trigonometric Functions

Trigonometric functions are the foundation of trigonometry and relate the angles of a triangle to the lengths of its sides. Understanding these functions is crucial to solving trigonometric problems. Functions like sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot) describe relationships within right-angled triangles but also define periodic waveforms useful in various scientific fields.

Each of these functions has specific properties and identities that express relationships between them. For instance, \( \text{csc} \text(x) \) is the reciprocal of \( \text{sin} \text(x) \), which can be vital in simplifying expressions and solving equations. Further, these functions are periodic, meaning they repeat their values at regular intervals, which is crucial when solving trigonometric equations.

Algebraic Confirmation

Algebraic confirmation is a formal method for proving that an equation is an identity. It involves manipulating and simplifying the equation using algebraic methods and trigonometric identities until both sides of the equation are the same. This process often includes combining like terms, converting all trigonometric functions to sines and cosines, and exploiting fundamental identities.

For instance, in the provided exercise, simplifying the expression involves using identities like \( \text{csc} \text(x) = \frac{1}{\text{sin} \text(x)} \) and \( \text{cot} \text(x) = \frac{\text{cos} \text(x)}{\text{sin} \text(x)} \) to rewrite and then combine terms. The goal is to show that after all simplifications, the left-hand side equals the right-hand side, therefore confirming the identity algebraically.

Cosecant (csc)

The cosecant function, denoted as \( \text{csc} \text(x) \), is one of the six primary trigonometric functions and is defined as the reciprocal of the sine function. In other words, \( \text{csc} \text(x) = \frac{1}{\text{sin} \text(x)} \). This means wherever the sine of an angle is zero, the cosecant of that angle is undefined, because division by zero is not possible.

The cosecant function is less commonly used than the sine function, but it is just as important when dealing with trigonometric identities and equations. In practice, to simplify complicated trigonometric equations involving the cosecant function, it can be helpful to rewrite it in terms of sine to allow for further simplification using more familiar sine-based trigonometric identities.