/*! 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 19 Verify the identity. $$\frac{\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 19

Verify the identity. $$\frac{\cot ^{2} t}{\csc t}=\frac{1-\sin ^{2} t}{\sin t}$$

Short Answer

Expert verified
The original identity \(\frac{\cot^{2} t}{\csc t} = \frac{1- \sin^{2} t}{\sin t}\) is valid as both sides simplify to \(\frac{\cos^{2} t}{\sin t}\)

Step by step solution

01

Rewrite the left side of the equation using the definitions of cotangent and cosecant

The cotangent of \(t\) is defined as the reciprocal of the tangent of \(t\), or \(\cot t = \frac{\cos t}{\sin t}\) and the cosecant of \(t\) is defined as the reciprocal of the sine of \(t\) or \(\csc t = \frac{1}{\sin t}\). Hence, substituted into the left side of our equation, it becomes \(\frac{(\frac{\cos t}{\sin t})^{2}}{\frac{1}{\sin t}} = \frac{\cos^{2} t}{\sin t}\)
02

Rewrite the right side of the equation using Pythagorean identity

The Pythagorean identity is \(\sin^{2} t + \cos^{2} t = 1\), which we can rearrange to get \(1 - \sin^{2} t = \cos^{2} t\). Substitute this into the right side of our equation gives \(\frac{\cos^{2} t}{\sin t}\)
03

Compare left side and right side of the equation

At this point, we see that the left side of the equation \(\frac{\cot^{2} t}{\csc t} = \frac{\cos^{2} t}{\sin t}\) is equal to the right side of the equation \(\frac{1- \sin^{2} t}{\sin t} = \frac{\cos^{2} t}{\sin t}\). Therefore, the original trigonometric identity is verified.

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

The displacement from equilibrium of a weight oscillating on the end of a spring is given by \(y=1.56 e^{-0.22 t} \cos 4.9 t,\) where \(y\) is the displacement (in feet) and \(t\) is the time (in seconds). Use a graphing utility to graph the displacement function for \(0 \leq t \leq 10\). Find the time beyond which the displacement does not exceed 1 foot from equilibrium.

Find the exact value of each expression. (a) \(\cos \left(\frac{\pi}{4}+\frac{\pi}{3}\right)\) (b) \(\cos \frac{\pi}{4}+\cos \frac{\pi}{3}\)

Use inverse functions where needed to find all solutions of the equation in the interval \([0,2 \pi)\). $$\sec ^{2} x-4 \sec x=0$$

Use inverse functions where needed to find all solutions of the equation in the interval \([0,2 \pi)\). $$\cot ^{2} x-9=0$$

Write the expression as the sine, cosine, or tangent of an angle. $$\cos \frac{\pi}{7} \cos \frac{\pi}{5}-\sin \frac{\pi}{7} \sin \frac{\pi}{5}$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.