/*! 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 Verify each of the trigonometric... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 23

Verify each of the trigonometric identities. $$ (\csc x+1)(\csc x-1)=\cot ^{2} x $$

Short Answer

Expert verified
The identity \((\csc x+1)(\csc x-1)=\cot ^{2} x\) is verified true.

Step by step solution

01

Rewrite the Left Side of the Equation

Let's expand the left side of the given identity \((\csc x + 1)(\csc x - 1)\). Use the difference of squares formula \((a+b)(a-b) = a^2 - b^2\).Thus, \(\csc^2 x - 1\).
02

Use Trigonometric Identity for \(\csc^2 x\)

Recall the Pythagorean identity:\[ \csc^2 x = 1 + \cot^2 x \]Substitute this into the expression from Step 1:\[ \csc^2 x - 1 = (1 + \cot^2 x) - 1 = \cot^2 x \]
03

Conclusion

We have shown that the left side of the original statement simplifies to \(\cot^2 x\), which is exactly what the right side of the equation is. Therefore, the 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!

Key Concepts

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

Difference of Squares
The difference of squares is a fundamental concept in algebra that can simplify many types of expressions. It takes the form
  • \((a+b)(a-b) = a^2 - b^2\)
This formula is used when you have the product of a sum and a difference of the same two terms. In our trigonometric problem, the terms are
  • \(\csc x + 1\) and \(\csc x - 1\)
Applying the formula here gives
  • \(\csc^2 x - 1\)
This effectively simplifies the multiplication of binomials into a single subtraction of squares. It's important to recognize this pattern as it reduces complex equations into simpler parts.
Pythagorean Identity
One of the most crucial identities in trigonometry is the Pythagorean identity. It provides a relationship between trigonometric functions and originates from the Pythagorean Theorem. The identity used here is:
  • \(\csc^2 x = 1 + \cot^2 x\)
This equation relates the cosecant and cotangent functions. In our problem, we use this identity to replace \(\csc^2 x - 1\) with \(\cot^2 x\). This simplification made it possible to verify the given identity. Understanding and applying the Pythagorean identity are key to solving trigonometric equations and simplifying expressions accurately.
Cosecant Function
The cosecant function, denoted as \(\csc x\), is the reciprocal of the sine function. It is defined as:
  • \(\csc x = \frac{1}{\sin x}\)
The cosecant function is crucial in various trigonometric identities and equations. In this exercise, \(\csc x\) is squared and used to simplify an expression using the difference of squares. Understanding what \(\csc x\) represents helps in recognizing how it interacts with other trigonometric functions, such as in the identity \(\csc^2 x = 1 + \cot^2 x\). This function mainly appears in identities, integrals, derivatives, and solving certain types of trigonometric equations.
Cotangent Function
The cotangent function, labeled as \(\cot x\), is another primary trigonometric function which is the reciprocal of tangent. It is expressed as:
  • \(\cot x = \frac{1}{\tan x}\)
  • Equivalently, \(\cot x = \frac{\cos x}{\sin x}\)
In trigonometric identities, such as \(\csc^2 x = 1 + \cot^2 x\), \(\cot x\) plays a significant role. It typically arises in situations where angles and slopes are analyzed within a right triangle. In our exercise, the identity simplifies to \(\cot^2 x\), demonstrating the cotangent function's importance in connecting various trigonometric properties. Understanding \(\cot x\) helps in simplifying expressions and solving real-world problems where angles and periodic patterns are involved.

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 the half-angle identities to find the exact values of the trigonometric expressions. $$ \sin \left(\frac{\pi}{8}\right) $$

Express \(\sin \left(\frac{x}{4}\right)\) in terms of the cosine of a single angle.

Two hikers leave from the same campsite and walk in different directions. The distance \(d\) in miles between the hikers can be found using the function \(d=\sqrt{13-12 \cos \theta}\), where \(\theta\) is the angle between the directions traveled by the hikers. Find a function for the distance between the hikers if \(\theta\) is doubled and then use a double-angle formula to write the function in terms of the cosine of a single angle \(\theta\).

Given \(\tan \left(\frac{A}{2}\right)=\pm \sqrt{\frac{1-\cos A}{1+\cos A}}\), verify that \(\tan \left(\frac{A}{2}\right)=\frac{\sin A}{1+\cos A}\). Substitute \(A=\pi\) into the identity and explain your results.

Verify the identities. $$ \sin ^{2}\left(\frac{x}{2}\right)=\frac{1-\sin \left(90^{\circ}-x\right)}{2} $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Calculus

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.