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Chapter 5: Problem 20

In Exercises I to \(42,\) verify each identity. $$\sec x=\frac{\cot x+\tan x}{\csc x}$$

Short Answer

Expert verified
The given trigonometric identity has been verified. The key was to convert the given identity in terms of sine and cosine functions, simplify the expression, and then use the Pythagorean identity.

Step by step solution

01

Rewrite the identity using fundamental trigonometric relationships

The given identity is \( \sec x = \frac{\cot x + \tan x}{\csc x} \). The fundamental trigonometric relationships can be expressed in terms of sine and cosine. So, let鈥檚 change the given identity in terms of sine and cosine: \( \sec x = \frac{\frac{1}{\tan x} + \tan x}{\frac{1}{\sin x}} \).
02

Simplify the expression

Simplify the above expression, \( \sec x = \frac{\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x}}{\frac{1}{\sin x}} \). This can be simplified to \( \sec x = \frac{\cos x + \sin^2 x}{\cos x} \).
03

Final verification

Now replace \( \sin^2 x \) with \( 1 - \cos^2 x \) to obtain \( \sec x = \frac{\cos x + 1 - \cos^2 x}{\cos x} \). Simplifying this equation gives \( \sec x = \frac{1}{\cos x} \), which is the definition of \( \sec x \). Thus, the identity has been verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Secant Function
The secant function, denoted as \( \sec x \), represents the reciprocal of the cosine function. Its definition is \( \sec x = \frac{1}{\cos x} \). As the cosecant function must avoid division by zero, \( \sec x \) is undefined whenever \( \cos x = 0 \).
This function becomes very useful when simplifying expressions or verifying identities involving cosine.

In the context of trigonometric identities, the secant function helps transform complex expressions by expressing terms in a reciprocal manner, facilitating stepwise simplification.
Cotangent Function
The cotangent function, or \( \cot x \), is the reciprocal of the tangent function. It can be expressed as \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \).
This function finds particular use in situations where sine and cosine expressions need to be divided.

Cotangent is undefined where \( \sin x = 0 \), as division by zero is not possible. Knowing the relationship between sine, cosine, and cotangent is crucial for simplifying expressions or verifying identities, especially when these trigonometric functions are involved.
Tangent Function
The tangent function, represented by \( \tan x \), is defined as the ratio of sine to cosine: \( \tan x = \frac{\sin x}{\cos x} \).
The tangent function is a critical concept in trigonometry, frequently appearing in various identities and applications.

One essential property of the tangent function is that it is undefined where \( \cos x = 0 \), which is when the angle is at odd multiples of \( \frac{\pi}{2} \). Understanding this function's properties allows you to manipulate and simplify trigonometric expressions effectively.
Cosecant Function
The cosecant function, indicated by \( \csc x \), is the reciprocal of the sine function, given by \( \csc x = \frac{1}{\sin x} \).
This reciprocal relationship makes it a fundamental component in many trigonometric identities and proofs.

Cosecant is undefined when \( \sin x = 0 \), making it essential to be cautious with angles at integer multiples of \( \pi \). Using the cosecant function can help simplify complex trigonometric expressions during manipulations or verifications.
Sine and Cosine Relationships
Sine and cosine are foundational elements of trigonometry. These functions form the basis for other trigonometric identities and relationships, like the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \).
Understanding the interplay between them aids in grasping more complex trigonometric functions such as secant, tangent, and their reciprocals.

Expressions involving sine and cosine can often be simplified by recognizing these fundamental relationships, reducing even the most seemingly intricate identities to more manageable forms.

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Most popular questions from this chapter

Solve tan \(\alpha=\left|\frac{-\sqrt{3}}{3}\right|\) for \(\alpha,\) where \(\alpha\) is an acute angle measured in degrees.

Verify the identity. $$\sec \left(\sin ^{-1} x\right)=\frac{\sqrt{1-x^{2}}}{1-x^{2}}$$

Use a graphing utility to graph equation. $$y=0.5 \sec ^{-1} \frac{x}{2}$$

Use a graphing utility to graph equation. $$y=\tan ^{-1}(x-1)$$

In Exercises 73 to \(88,\) verify the identity. $$\cos \left(\frac{\pi}{2}-\theta\right)=\sin \theta$$
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