/*! 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 46 Verify the identity. $$\sec ^{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 46

Verify the identity. $$\sec ^{2}\left(\frac{\pi}{2}-x\right)-1=\cot ^{2} x$$

Short Answer

Expert verified
The left-hand side of the given equation simplifies to \( \cot^2(x) \), which matches the right-hand side of the equation. Therefore, the identity is verified.

Step by step solution

01

Recognize Co-function Identity

To simplify the left hand side of the equation, recognize that \( \sec(\frac \pi 2 - x) \) is equivalent to \( \csc(x) \), the co-function identity of \(\sec\). So, \(\sec ^{2}\left(\frac{\pi}{2}-x\right) - 1\) can be rewritten as \( \csc^2(x) - 1\).
02

Use Pythagorean Identity

Now utilize the Pythagorean identity \( \csc^2(x) = 1 + \cot^2(x) \). By substituting this into the equation, we have \( 1 + \cot^2(x) - 1 \).
03

Simplify Expression

Simplify the expression to get \( \cot^2(x) \).
04

Verify the Identity

This matches the right-hand side of the equation. Thus, the identity \( \sec ^{2}(\frac{\pi}{2}-x)-1=\cot ^{2} x \) 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

Use inverse functions where needed to find all solutions of the equation in the interval \([0,2 \pi)\). $$\sec ^{2} x+\tan x-3=0$$

Solve the multiple-angle equation. $$\cos \frac{x}{2}=\frac{\sqrt{2}}{2}$$

Use the Quadratic Formula to solve the equation in the interval \([0,2 \pi)\). Then use a graphing utility to approximate the angle \(x\). $$\tan ^{2} x+3 \tan x+1=0$$

Find the exact value of the expression. $$\cos 120^{\circ} \cos 30^{\circ}+\sin 120^{\circ} \sin 30^{\circ}$$

Find the exact values of the sine, cosine, and tangent of the angle. $$-\frac{13 \pi}{12}$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Applied 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.