/*! 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 39 Verify the identity. $$\sin ^{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 39

Verify the identity. $$\sin ^{1 / 2} x \cos x-\sin ^{5 / 2} x \cos x=\cos ^{3} x \sqrt{\sin x}$$

Short Answer

Expert verified
The given identity \(\sin ^{1 / 2} x \cos x - \sin ^{5 / 2} x \cos x = \cos ^{3} x \sqrt{\sin x}\) holds if \(\cos x = 1\). This will be the case when \(x=2n\pi\), \(n\) being an integer.

Step by step solution

01

Rewrite the Equation

Rewrite the equation in a format in terms of sine. Remembering that \(\cos^{2} x = 1 - \sin^{2} x\), this allows the right-hand side (RHS) to be rewritten in terms of \(\sin x\) as, \(\cos^{3}x \sqrt{\sin x}=(1-\sin^{2} x)\sqrt{\sin x}\). Using the distributive property to expand yields, \( \sqrt{\sin x}-\sin^{2} x \sqrt{\sin x}\).
02

Simplify the Right Hand Equation

\(\sqrt{\sin x} - \sin^{2} x \sqrt{\sin x}\) can be simplified to \(\sin ^{1 / 2} x - \sin ^{2.5} x\). The expression \(\sin ^{2.5} x\) is the same as \(\sin^{5 / 2} x\). So this expression simplifies to \(\sin^{1 / 2} x - \sin^{5 / 2} x\).
03

Match the Left and Right Sides

Comparing this with the left-hand side (LHS), \(\sin^{1 / 2} x \cos x - \sin^{5 / 2} x \cos x\), notice that both expressions match if we consider \(\cos x = 1\).

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 values of \(\sin (u / 2), \cos (u / 2),\) and \(\tan (u / 2)\) using the half-angle formulas. $$\sin u=\frac{5}{13}, \quad \pi / 2

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\sec \frac{u}{2}=\pm \sqrt{\frac{2 \tan u}{\tan u+\sin u}}$$

Use the half-angle formulas to simplify the expression. $$-\sqrt{\frac{1-\cos 8 x}{1+\cos 8 x}}$$

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$165^{\circ}$$

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\frac{\sin x+\sin y}{\cos x-\cos y}=-\cot \frac{x-y}{2}$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Statistics

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.