/*! 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 80 Perform the addition or subtract... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 80

Perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer. $$\tan x-\frac{\sec ^{2} x}{\tan x}$$

Short Answer

Expert verified
Our simplified answer for the expression is \(-\cot x\).

Step by step solution

01

Express sec in terms of tan

Express \(\sec ^{2} x\) in terms of \(\tan^{-1}\) because sec and tan are co-function. We know that the relationship between tan and sec is given by the Pythagorean identity \(\tan^{2} x = \sec^{2} x - 1\). From this identity, we get \(\sec ^{2} x = \tan^{2} x + 1\).
02

Substituting the value of sec

Substitute the value of \(\sec ^{2} x\) that we found from step 1 into the original equation, \(\tan x - \frac{\sec^{2} x}{\tan x}\) becomes \(\tan x - \frac{\tan^{2} x + 1}{\tan x}\).
03

Split the fraction and simplify

Split the fraction \(\frac{\tan^{2} x +1}{\tan x}\) and simplify as follows: \(\tan x - (\tan x + \frac{1}{\tan x})\)
04

Cancel out common terms

Finally, we simplify the expression by canceling the common terms, the 'tan x' in both the first part and the first part of the subtraction cancels out and leaves us with \(-\frac{1}{\tan x}\). We also know that \(-\frac{1}{\tan x} = -\cot x\). So, the expression simplifies to \(-\cot x\).

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!

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

Write the expression as the sine, cosine, or tangent of an angle. $$\frac{\tan 140^{\circ}-\tan 60^{\circ}}{1+\tan 140^{\circ} \tan 60^{\circ}}$$

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{13 \pi}{12}$$

Find the exact values of the sine, cosine, and tangent of the angle. $$-105^{\circ}$$

The monthly sales \(S\) (in thousands of units) of a seasonal product are approximated by $$S=74.50+43.75 \sin \frac{\pi t}{6}$$ where \(t\) is the time (in months), with \(t=1\) corresponding to January. Determine the months in which sales exceed 100,000 units.

Use the Quadratic Formula to solve the equation in the interval \([0,2 \pi)\). Then use a graphing utility to approximate the angle \(x\). $$\tan ^{2} x+3 \tan x+1=0$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Decision 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.