/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 31 Verify each identity. $$\frac{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 31

Verify each identity. $$\frac{\cos x}{1-\sin x}+\frac{1-\sin x}{\cos x}=2 \sec x$$

Short Answer

Expert verified
After going through the steps, it is verified that \( \frac{\cos x}{1-\sin x}+\frac{1-\sin x}{\cos x}\) indeed equals \(2 \sec x\). Therefore, the given identity is correct.

Step by step solution

01

Simplify the LHS by Adding the Two Fractions

On the left-hand side, we have \(\frac{\cos x}{1-\sin x} + \frac{1-\sin x}{\cos x}\). To add these two fractions, we can find a common denominator, which will be \((1-\sin x) \cdot \cos x\). After doing so, the numerator becomes \(\cos^2 x + (1- \sin x)^2\)
02

Simplify the Numerator by Expanding

The numerator \(\cos^2 x + (1- \sin x)^2\) can be expanded to become \(\cos^2 x + 1 - 2 \sin x + \sin^2 x\). The term \(\cos^2 x + \sin^2 x\) in the numerator can be replaced by 1 (due to the Pythagorean identity \(\sin^2 x+ \cos^2 x = 1\)) and then simplifies to \(2- 2 \sin x\)
03

Simplify the Denominator

We know from step 1 that the denominator is \((1 - \sin x) \cdot \cos x\). It remains as is.
04

Express in Terms of Secant

Now our left-hand side is \(\frac{2 - 2 \sin x}{(1 - \sin x) \cdot \cos x}\). This can further be simplified to \(2 \cdot \frac{1 - \sin x}{(1 - \sin x) \cdot \cos x}\), which equals \(2 \cdot \sec x\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Pythagorean Identity
The Pythagorean Identity is one of the fundamental building blocks in trigonometry. It's expressed as \( \sin^2 x + \cos^2 x = 1 \). This identity arises because the square of the sine and cosine of an angle will always add up to one, influenced by the intrinsic properties of right triangles and the unit circle.

In the given solution, the identity is used to simplify the expression \( \cos^2 x + (1 - \sin x)^2 \). Here's how it works:
  • Start by expanding \((1 - \sin x)^2\) into \(1 - 2 \sin x + \sin^2 x\).
  • Combine this with \(\cos^2 x\) to get \(\cos^2 x + \sin^2 x + 1 - 2 \sin x\).
  • Thanks to the Pythagorean Identity, \( \cos^2 x + \sin^2 x = 1\) simplifies the expression to \(2 - 2 \sin x\).
This simplification is crucial as it allows for further reduction and manipulation of trigonometric expressions. Always remember this identity when working with sine and cosine, as it can greatly simplify your solution process.
Fraction Addition
Adding fractions is a key mathematical operation that often crops up in algebra and calculus. To add fractions, it's essential to find a common denominator. This makes the addition process straightforward and helps avoid errors.

In the exercise we're solving, the addition of fractions \(\frac{\cos x}{1-\sin x} + \frac{1-\sin x}{\cos x}\) requires a common denominator: \((1-\sin x) \cdot \cos x\). Here's the step-by-step to find the sum:
  • Multiply the numerator and denominator of each fraction by what the other fraction is missing. For \(\frac{\cos x}{1-\sin x}\), multiply by \(\cos x \), and for \(\frac{1-\sin x}{\cos x}\), multiply by \(1-\sin x\).
  • This gives both fractions the same denominator, allowing them to be combined into a single fraction: \(\frac{\cos^2 x + (1- \sin x)^2}{(1-\sin x) \cdot \cos x}\).
Finding a common denominator is crucial in fraction addition, especially in trigonometric expressions, enabling further simplification and resolution of the equation.
Trigonometric Simplification
Trigonometric simplification involves reducing complex trigonometric expressions using basic identities and algebraic manipulations. It's widely used to verify identities, solve equations, or integrate functions.

In this problem, simplification starts from the initial combined fraction, \(\frac{\cos^2 x + (1- \sin x)^2}{(1-\sin x) \cdot \cos x}\), which highlights several techniques:
  • Apply identities like the Pythagorean identity \(\cos^2 x + \sin^2 x = 1\) to simplify expressions.
  • Factor expressions such as transforming \(2 - 2 \sin x\) into \(2(1 - \sin x)\).
  • Cancel terms tactically, here dividing by \( (1 - \sin x) \) to reduce the expression to \(2 \sec x\).
Trigonometric simplification leverages well-known identities and algebra tricks to make sense of complex trigonometric equations. By mastering this skill, complex problems can be broken down into manageable parts.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises \(97-116,\) use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$3 \cos x-6 \sqrt{3}=\cos x-5 \sqrt{3}$$

In Exercises \(97-116,\) use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$2 \cos 2 x+1=0$$

In Exercises \(121-126,\) solve each equation on the interval \([0,2 \pi)\) $$|\cos x|=\frac{\sqrt{3}}{2}$$

In Exercises \(97-116,\) use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$\cos x \csc x=2 \cos x$$

Use words to describe the formula for: Without showing algebraic details, describe in words how to reduce the power of \(\cos ^{4} x\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.