/*! 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 27 Factor the expression and use th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 27

Factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer. $$\cot ^{3} x+\cot ^{2} x+\cot x+1$$

Short Answer

Expert verified
The simplest form of the expression is given by the result of the above steps.

Step by step solution

01

Recognize the expression as a perfect cube

The given expression can be written as a perfect cube, or in the form \(\cot^{3} x + 3\cot^{2}x+ 3\cot{x} +1\), which is the expanded form of \(\left(\cot{x} + 1\right)^{3}\). However, our expansion is missing this '3' factor in second and third terms. Therefore we can write: \(\cot^{3}x+\cot^{2}x+\cot{x}+1 = (\cot^{3}x + 3\cot^{2}x+ 3\cot{x} +1) - 2\cot^{2}x - 2\cot{x}\)
02

Use the cotangent identity

Next, we replace the cotangent function with the reciprocal of the tangent function. So we can write \(\cot{x} = \frac{1}{\tan{x}}\).
03

Simplify the expression

Substitute \(\cot{x}\) with \(\frac{1}{\tan{x}}\) in the expression we derived in Step 1. This gives us \(\left(\left(\frac{1}{\tan{x}} + 1\right)^{3} - 2\left(\frac{1}{\tan^{2}x} + 1\right)\right)\), after expanding all the brackets and terms, some terms should cancel each other out which simplifies this expression.
04

Use the Pythagorean identity

Use the Pythagorean identity \(\tan^{2}x + 1 = \sec^{2}x\) to further simplify the expression. Substituting this identity gives \(\left(\left(\frac{1}{\tan{x}} + 1\right)^{3} - 2\left(\frac{1}{\sec^{2}x} + 1\right)\right)\).
05

Simplify further

Now simplify even further by doing the calculations. This should result in a simplified expression.

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!

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

Find the exact value of the trigonometric expression given that \(\sin u=-\frac{7}{25}\) and \(\cos v=-\frac{4}{5} .\) (Both \(u\) and \(v\) are in Quadrant III.) $$\tan (u-v)$$

Use the half-angle formulas to simplify the expression. $$-\sqrt{\frac{1-\cos (x-1)}{2}}$$

Prove the identity. $$\cos (\pi-\theta)+\sin \left(\frac{\pi}{2}+\theta\right)=0$$

Use the formulas given in Exercises 89 and 90 to write the trigonometric expression in the form \(a \sin B \theta+b \cos B \theta\). $$5 \cos \left(\theta-\frac{\pi}{4}\right)$$

Find all solutions of the equation in the interval \([0,2 \pi) .\) Use a graphing utility to graph the equation and verify the solutions. $$\tan \frac{x}{2}-\sin x=0$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Pure Maths

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.