/*! 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 68 Show that \(\frac{\cos x}{1-\sin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 68

Show that \(\frac{\cos x}{1-\sin ^{2} x}=\sec x\) is an identity.

Short Answer

Expert verified
By using the basic properties of trigonometric functions and the Pythagorean identity, we have shown that \(\frac{\cos x}{1-\sin^{2}x}=\sec x\) is indeed an identity.

Step by step solution

01

Understand the basics

Let's begin by understanding the basic properties of trigonometric functions. Specifically, remember that \(\sec x = \frac{1}{\cos x}\) and \(\sin^2x + \cos^2x = 1\).
02

Substitute for \(sec x\)

We start by replacing secant on the right-hand side by its equivalent in terms of cosine, so \(\sec x = \frac{1}{\cos x}\). So now the equation we want to prove becomes \(\frac{\cos x}{1-\sin^{2}x}=\frac{1}{\cos x}\).
03

Cross-multiply

You then cross-multiply each side by \(\cos x\) and \(1-\sin^{2}x\). This simplifies the equation to \(\cos^{2}x = 1 - \sin^{2}x\).
04

Use the Pythagorean identity

Next, substitute the Pythagorean identity for the right side of the equation. Remember that \(\sin^{2}x + \cos^{2}x = 1\), which rearranges to \(\cos^{2}x = 1 - \sin^{2}x\). So substituting, we get \(\cos^{2}x = \cos^{2}x\).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Function
The cosine function, often denoted as \( \cos x \), is a fundamental part of trigonometry. It expresses the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Unlike the sine function, which deals with the opposite side, cosine helps us understand and analyze angles based on the adjacent side.
  • Cosine values range from -1 to 1.
  • The cosine function is periodic, with a period of \(2\pi\).
  • It is an even function, meaning \(\cos(-x) = \cos(x)\).
Understanding the cosine function is crucial in solving trigonometric identities and equations. In our exercise, the cosine function appears in both the numerator and denominator, and understanding how it behaves is key to manipulating the equation.
Secant Function
The secant function, represented as \(\sec x\), is the reciprocal of the cosine function. This means \(\sec x = \frac{1}{\cos x}\).
  • Since it is a reciprocal, secant is undefined where cosine is zero because division by zero is undefined.
  • Secant values can range from 1 and above or -1 and below, reflecting the reciprocal nature of cosine's values.
  • It is particularly useful in calculus and physics, where division by cosine frequently occurs.
In proving the identity \(\frac{\cos x}{1 - \sin^2 x} = \sec x\), converting secant to its cosine equivalent helps simplify the equation. Recognizing this relationship is fundamental when working with trigonometric identities.
Pythagorean Identity
The Pythagorean identity is one of the most useful identities in trigonometry, usually represented as \(\sin^2 x + \cos^2 x = 1\). This identity is derived from the equation of a unit circle, where any point \((\cos x, \sin x)\) satisfies the equation \(x^2 + y^2 = 1\).
  • The identity shows the inherent relationship between sine and cosine for any angle \(x\).
  • You can rearrange it to find expressions for \(\cos^2 x\) or \(\sin^2 x\), depending on what is needed.
  • In our exercise, it is rewritten as \(\cos^2 x = 1 - \sin^2 x\), simplifying the original equation to a true statement.
Using the Pythagorean identity allowed us to confirm \(\cos x = 1 - \sin^2 x\), showing how essential it is in proving that trigonometric identities hold true.

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

Simplify each expression. $$ \sin ^{2} \frac{\theta}{2}-\cos ^{2} \frac{\theta}{2} $$

a. open-Ended Sketch a triangle. Specify three of its measures so that you can use the Law of Cosines to find the remaining measures. b. Solve for the remaining measures of the triangle.

Given \(\cos \theta=\frac{3}{5} \operatorname{and} 270^{\circ}<\theta<360^{\circ},\) find the exact value of each expression. $$ \cos 2 \theta $$

Geometry The lengths of the sides of a triangle are \(7.6 \mathrm{cm}, 8.2 \mathrm{cm},\) and 5.2 \(\mathrm{cm} .\) Find the measure of the largest angle.

\(\ln \triangle D E F, d=20 \mathrm{ft}, e=25 \mathrm{ft},\) and \(m \angle F=98^{\circ} .\) Find \(m \angle D\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

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.