/*! 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 23 Establish each identity. $$1+\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 23

Establish each identity. $$1+\tan ^{2}(-\theta)=\sec ^{2} \theta$$

Short Answer

Expert verified
left(\theta\right)

Step by step solution

01

Understand the given identity

Identify the given trigonometric identity: o The identity to establish is: o However, using the even function property in trigonometry which states that o Now rewrite the left side as: o
02

Use the Pythagorean identity

Recall that one of the fundamental Pythagorean identities in trigonometry is: o Use this Pythagorean identity to transform the left side from Step 1.
03

Simplify the expression

Substitute o We establish that the left side to the right side.

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.

Pythagorean Identity
The Pythagorean identity is fundamental in trigonometry and is often used to simplify and prove various expressions. The most commonly used Pythagorean identity is: \(\text{sin}^2 \theta + \text{cos}^2 \theta = 1\). Using this identity, we can derive other useful forms, such as: \(\text{tan}^2 \theta + 1 = \text{sec}^2 \theta\). This identity is used in our given problem to transform expressions involving tangent and secant. Understanding these relationships is crucial for manipulating and simplifying trigonometric expressions.
Even Function Property
An important property in trigonometry is that some functions are even, meaning they are symmetrical about the y-axis. For the context of our identity, the cosine function \(\text{cos}(-\theta) = \text{cos}(\theta)\) is an even function. Because secant is the reciprocal of cosine, \(\text{sec}(-\theta) = \text{sec}(\theta)\) is also true. Recognizing this property allows us to simplify the given problem by noting that \(1 + \text{tan}^2(-\theta) = 1 + \text{tan}^2(\theta)\), since tangent squared (\(\text{tan}^2\)) eliminates the effect of the angle being negative.
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions often requires using known identities and properties to transform complex expressions into simpler, equivalent forms. In the given exercise, we start from \(1 + \text{tan}^2(-\theta)\). By using the even function property, we get \(1 + \text{tan}^2 \theta\). Then, applying the Pythagorean identity, we see that \(1 + \text{tan}^2 \theta = \text{sec}^2 \theta\). This step-by-step process demonstrates how combining various trigonometric rules can simplify and establish identities.
Tangent and Secant Functions
The tangent and secant functions are closely related through the Pythagorean identity. The tangent function is defined as \(\text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta}\), and the secant function is \(\text{sec} \theta = \frac{1}{\text{cos} \theta}\). By squaring the tangent function and adding one, we get \(1 + \text{tan}^2 \theta = \frac{1}{\text{cos}^2 \theta}\), which is exactly \(\text{sec}^2 \theta\). Understanding these relationships allows us to transform expressions like in our given problem and recognize underlying identities between these functions.

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

Establish each identity. $$ \sin (\theta+k \pi)=(-1)^{k} \sin \theta, k \text { any integer } $$

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Write \(f(x)=\frac{1}{4} x^{2}+x-2\) in vertex form.

Find the exact value of each expression. $$ \sin \left[\sin ^{-1} \frac{3}{5}-\cos ^{-1}\left(-\frac{4}{5}\right)\right] $$

Calculus Show that the difference quotient for \(f(x)=\cos x\) is given by $$ \begin{aligned} \frac{f(x+h)-f(x)}{h} &=\frac{\cos (x+h)-\cos x}{h} \\ &=-\sin x \cdot \frac{\sin h}{h}-\cos x \cdot \frac{1-\cos h}{h} \end{aligned} $$

Geometry: Angle between Two Lines Let \(L_{1}\) and \(L_{2}\) denote two nonvertical intersecting lines, and let \(\theta\) denote the acute angle between \(L_{1}\) and \(L_{2}\) (see the figure). Show that $$ \tan \theta=\frac{m_{2}-m_{1}}{1+m_{1} m_{2}} $$ where \(m_{1}\) and \(m_{2}\) are the slopes of \(L_{1}\) and \(L_{2},\) respectively. [Hint: Use the facts that \(\tan \theta_{1}=m_{1}\) and \(\left.\tan \theta_{2}=m_{2} .\right]\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Discrete 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.