/*! 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 116 Verify the identity: $$\sin 2 ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 116

Verify the identity: $$\sin 2 x=\frac{2 \tan x}{1+\tan ^{2} x}$$ (Section \(5.3,\) Examples 3 and 6 )

Short Answer

Expert verified
The identity \(\sin 2 x=\frac{2 \tan x}{1+\tan ^{2} x}\) is true, as the last simplification step yields \(\sin 2 x\) for the right-hand side.

Step by step solution

01

Write the Expressions

Start off by writing down the two sides of the identity we are trying to prove: sin(2x) and (2tan(x)) / (1 + tan^2(x)).
02

Use the Formula for Sin(2x)

Recall that sin(2x) = 2sin(x)cos(x). Substitute this into the left-hand side of the equation.
03

Express Tan(x) in Terms of Sine and Cosine

Remember that tan(x) = sin(x) / cos(x). Replace each tan(x) on the right-hand side of the equation by sin(x) / cos(x).
04

Perform the Substitution

Now substitute tan(x) = sin(x) / cos(x) into (2tan(x)) / (1 + tan^2(x)) to get (2sin(x) / cos(x)) / (1 + (sin^2(x) / cos^2(x))).
05

Simplify the Result

You have to simplify the fraction. To do so, multiply the numerator and the denominator by cos^2(x) to yield (2sin(x)cos(x)) / (cos^2(x) + sin^2(x)). Now, use the Pythagorean identity sin^2(x) + cos^2(x) = 1 and simplify to 2sin(x)cos(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

Use the vectors $$\mathbf{u}=a_{1} \mathbf{i}+b_{1} \mathbf{j}, \quad \mathbf{v}=a_{2} \mathbf{i}+b_{2} \mathbf{j}, \quad \text { and } \quad \mathbf{w}=a_{3} \mathbf{i}+b_{3} \mathbf{j},$$ to prove the given property. $$(c \mathbf{u}) \cdot \mathbf{v}=c(\mathbf{u} \cdot \mathbf{v})$$

Describe how to find the angle between two vectors.

Help you prepare for the material covered in the first section of the next chapter. Graph \(x+2 y=2\) and \(x-2 y=6\) in the same rectangular coordinate system. At what point do the graphs intersect?

A wagon is pulled along level ground by exerting a force of 40 pounds on a handle that makes an angle of \(32^{\circ}\) with the horizontal. How much work is done pulling the wagon 100 feet? Round to the nearest foot-pound.

Two buildings of equal height are 800 feet apart. An observer on the street between the buildings measures the angles of elevation to the tops of the buildings as \(27^{\circ}\) and \(41^{\circ} .\) How high, to the nearest foot, are the buildings?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

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.