Problem 34
Question
You can determine whether or not an equation may be a trigonometric identity by graphing the expressions on either side of the equals sign as two separate functions. If the graphs do not match, then the equation is not an identity. If the two graphs do coincide, the equation might be an identity. The equation has to be verified algebraically to ensure that it is an identity. $$ \frac{\sec ^{2} x}{\tan x}=\sec x \csc x $$
Step-by-Step Solution
Verified Answer
Yes, \( \frac{\sec^2 x}{\tan x} = \sec x \csc x \) is a trigonometric identity.
1Step 1: Understand the Equation
We are given \( \frac{\sec^2 x}{\tan x} = \sec x \csc x \). Our task is to graph both sides of the equation as individual functions.
2Step 2: Graph the Left Side
The left side is \( \frac{\sec^2 x}{\tan x} \). We need to graph this function, which involves knowing that \( \sec x = \frac{1}{\cos x} \) and \( \tan x = \frac{\sin x}{\cos x} \). Thus, the left side becomes \( \frac{1/\cos^2 x}{\sin x/\cos x} = \frac{1}{\cos x \sin x} \).
3Step 3: Graph the Right Side
The Right side is \( \sec x \csc x \) which can also be written using \( \sec x = \frac{1}{\cos x} \) and \( \csc x = \frac{1}{\sin x} \). Therefore, the right side becomes \( \frac{1}{\cos x \sin x} \).
4Step 4: Compare Graphs
Plot the graphs of \( \frac{1}{\cos x \sin x} \) as derived from both sides using a graphing calculator or software. Check if they coincide.
5Step 5: Algebraically Verify the Identity
Since the graphs coincide, rewrite both sides. Both simplify to \( \frac{1}{\cos x \sin x} \), confirming they are indeed algebraically equal.
Key Concepts
Graphing FunctionsSecant and CosecantAlgebraic VerificationTangent and SecantFunction Comparison
Graphing Functions
Before delving into the details of trigonometric identities, it's essential to understand the concept of graphing functions. Graphing helps visualize mathematical expressions and ascertain whether an equation might be an identity. When you graph two functions, you're essentially plotting their values over a range of inputs. If their graphs look identical, further verification is needed to confirm the identity.
It's like drawing two pictures and comparing them: if they appear the same at first glance, you double-check to ensure they truly match. Graphing gives a sneak peek into the behavior of functions, setting the stage for deeper exploration.
Using graphing software or calculators is a practical way to visualize functions' behaviors and compare their outputs. Remember, just because graphs overlap, doesn't always mean the equation is an identity, it just means further checking is warranted.
It's like drawing two pictures and comparing them: if they appear the same at first glance, you double-check to ensure they truly match. Graphing gives a sneak peek into the behavior of functions, setting the stage for deeper exploration.
Using graphing software or calculators is a practical way to visualize functions' behaviors and compare their outputs. Remember, just because graphs overlap, doesn't always mean the equation is an identity, it just means further checking is warranted.
Secant and Cosecant
The secant (\( \sec x \)) and cosecant (\( \csc x \)) functions are reciprocal trigonometric functions with essential roles in verifying the equation \( \frac{\sec^2 x}{\tan x} = \sec x \csc x \). Understanding these functions aids in graphing and simplifying expressions.
The secant function is defined as the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \). This means wherever the cosine function is zero, the secant function will be undefined, resulting in vertical asymptotes in the graph.
Similarly, the cosecant function is the reciprocal of the sine function: \( \csc x = \frac{1}{\sin x} \). This means the cosecant function will be undefined wherever sine is zero. Understanding these definitions is crucial as they influence how the graphs behave and are used in simplifying trigonometric identities.
The secant function is defined as the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \). This means wherever the cosine function is zero, the secant function will be undefined, resulting in vertical asymptotes in the graph.
Similarly, the cosecant function is the reciprocal of the sine function: \( \csc x = \frac{1}{\sin x} \). This means the cosecant function will be undefined wherever sine is zero. Understanding these definitions is crucial as they influence how the graphs behave and are used in simplifying trigonometric identities.
Algebraic Verification
Verifying an identity algebraically involves manipulating one or both sides of the equation to prove their equivalence. This verification is crucial even when the graphs coincide. It's like checking the math behind a beautiful picture to ensure it's accurate.
For our given equation, simplify each side algebraically. For the left side: \( \frac{\sec^2 x}{\tan x} = \frac{1/\cos^2 x}{\sin x /\cos x} \) simplifies to \( \frac{1}{\cos x \sin x} \).
The right side, \( \sec x \csc x = \frac{1}{\cos x \sin x} \) , confirms this same result. If both sides simplify to the same expression, as we see here, the identity stands verified algebraically.
For our given equation, simplify each side algebraically. For the left side: \( \frac{\sec^2 x}{\tan x} = \frac{1/\cos^2 x}{\sin x /\cos x} \) simplifies to \( \frac{1}{\cos x \sin x} \).
The right side, \( \sec x \csc x = \frac{1}{\cos x \sin x} \) , confirms this same result. If both sides simplify to the same expression, as we see here, the identity stands verified algebraically.
Tangent and Secant
Understanding the tangent (\( \tan x \)) and its relationship with the secant (\( \sec x \)) is critical. The tangent function is defined as the ratio of sine to cosine: \( \tan x = \frac{\sin x}{\cos x} \). This relationship can lead to transformations, helping with algebraic simplifications in identities.
The given equation \( \frac{\sec^2 x}{\tan x} \) begins with these function definitions. By substituting secant and tangent meaningfully, it simplifies complex expressions. The interplay between tangent and secant functions often resolves into simpler expressions during identity verification, reinforcing the power of understanding basic trigonometric relations.
The given equation \( \frac{\sec^2 x}{\tan x} \) begins with these function definitions. By substituting secant and tangent meaningfully, it simplifies complex expressions. The interplay between tangent and secant functions often resolves into simpler expressions during identity verification, reinforcing the power of understanding basic trigonometric relations.
Function Comparison
Function comparison plays an integral role in determining whether an equation stands as a trigonometric identity. Once both sides of the given equation are graphed, comparing these functions gives an initial clue about their possible equivalence.
In comparing the functions \( \frac{\sec^2 x}{\tan x} \) and \( \sec x \csc x \), you start by visually ensuring the range and positioning of each graph line up perfectly.
In comparing the functions \( \frac{\sec^2 x}{\tan x} \) and \( \sec x \csc x \), you start by visually ensuring the range and positioning of each graph line up perfectly.
- If differences appear, the equation may not be an identity.
- If graphs overlap completely, digging deeper with algebraic checks becomes necessary.
Other exercises in this chapter
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