/*! 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 17 Verify each identity. $$\sin t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 17

Verify each identity. $$\sin t \tan t=\frac{1-\cos ^{2} t}{\cos t}$$

Short Answer

Expert verified
The given trigonometric identity, \(\sin t \tan t = \frac{1-\cos ^{2} t}{\cos t}\), is verified as correct by applying the substitution method and Pythagorean trigonometric identity.

Step by step solution

01

Recognize Trig Identities

By recognizing the identities, notice that \(\tan t\) can be written as the ratio of \(\sin t\) to \(\cos t\). Hence, substitute \(\tan t\) with \(\frac{\sin t}{\cos t}\) on the left side of the equation.
02

Simplify the Expression

After the substitution, the left side becomes \(\sin t \cdot \frac{\sin t}{\cos t}\). By reducing the expression, it simplifies to \(\frac{\sin^{2} t}{\cos t}\).
03

Apply Trig Identity

Apply the Pythagorean identity which states that \(\sin^2 t + \cos^2 t = 1\). By rearranging the formula, \(\sin^2 t\) can be expressed as \(1 - \cos^2 t\). Substitute \(\sin^2 t\) with \(1 - \cos^2 t\) in the equation.
04

Verify the Identity

Once the substitution is done, the left side of the equation becomes \(\frac{1-\cos^{2} t}{\cos t}\) which exactly the same as the right side of the given identity, thus proving the identity as correct.

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 a graphing utility to approximate the solutions of each equation in the interval \([0,2 \pi) .\) Round to the nearest hundredth of a radian. $$15 \cos ^{2} x+7 \cos x-2=0$$

Describe a natural periodic phenomenon. Give an example of a question that can be answered by a trigonometric equation in the study of this phenomenon.

Verify each identity. $$\frac{\sin ^{3} x-\cos ^{3} x}{\sin x-\cos x}=1+\sin x \cos x$$

Use a graphing utility to approximate the solutions of each equation in the interval \([0,2 \pi) .\) Round to the nearest hundredth of a radian. $$\cos x=x$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning.After using an identity to determine the exact value of \(\sin 105^{\circ}, 1\) verified the result with a calculator.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Geometry

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.