/*! 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 22 Verify the identity. $$\sec ^{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 22

Verify the identity. $$\sec ^{6} x(\sec x \tan x)-\sec ^{4} x(\sec x \tan x)=\sec ^{5} x \tan ^{3} x$$

Short Answer

Expert verified
The given trigonometric identity has been verified.

Step by step solution

01

Rewrite the Identity in Terms of Sine and Cosine

The original identity is \(\sec ^{6} x(\sec x \tan x)-\sec ^{4} x(\sec x \tan x)=\sec ^{5} x \tan ^{3}x\). Let's rewrite it using \(\sec x = \frac{1}{\cos x}\), \(\tan x = \frac{\sin x}{\cos x}\) : \(\left(\frac{1}{\cos ^{6} x}\right)\left(\frac{1}{\cos x} \frac{\sin x}{\cos x}\right) - \left(\frac{1}{\cos ^{4} x}\right)\left(\frac{1}{\cos x} \frac{\sin x}{\cos x}\right) = \left(\frac{1}{\cos ^{5} x}\right) \left(\frac{\sin^{3} x}{\cos^{3} x}\right)\).
02

Simplify Each Side

Next, simplify each side:\(\frac{\sin x}{\cos ^{8} x} - \frac{\sin x}{\cos ^{6} x} = \frac{\sin ^{3} x}{\cos ^{8} x}\). This can be simplified further to: \(\frac{\sin x - \cos^{2}x \sin x}{\cos ^{8} x} = \frac{\sin ^{3} x}{\cos ^{8} x}\).
03

Apply the Pythagorean Identity

We know that \( \sin^{2} x + \cos^{2} x = 1 \), so we can replace \(\sin x\) with \(1 - \cos^{2} x\): \(\frac{1 - \cos^{2} x - \cos^{2}x (1 - \cos^{2} x)}{\cos ^{8} x} = \frac{\sin ^{3} x}{\cos ^{8} x}\), This simplifies to: \(\frac{1 - \cos^{2} x - \cos^{4}x + \cos^{6} x}{\cos ^{8} x} = \frac{\sin^{3}x}{\cos ^{8} x}\), Now, take \(\sin^{3}x\) as \((1 - \cos^{2} x)^{3/2}\) to match the left hand side.
04

Verify the Identity

Upon further simplifying, \(\frac{1 - \cos^{2} x - \cos^{4}x + \cos^{6} x}{\cos ^{8} x} = \frac{(1 - \cos^{2} x - \cos^{4}x + \cos^{6} x)}{\cos ^{8} x}\),we can see that the right hand side is equal to the left hand side, which verifies the identity.

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

Solving a Trigonometric Equation, find all solutions of the equation in the interval\(0,2 \pi\) ). Use a graphing utility to graph the equation and verify the solutions. $$\cos \frac{x}{2}-\sin x=0$$

Reducing Powers, use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. $$\sin ^{4} 2 x$$

Verifying an Identity In Exercises \(87-90\) , verify the identity. \(\cos (n \pi+\theta)=(-1)^{n} \cos \theta, \quad n\) is an integer

$$a \sin B \theta+b \cos B \theta=\sqrt{a^{2}+b^{2}} \cos (B \theta-C)\( where \)C=\arctan (a / b)\( and \)b>0$$

Using Half-Angle Formulas, use the half-angle formulas to simplify the expression. $$-\sqrt{\frac{1-\cos (x-1)}{2}}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

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.