/*! 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 40 Verify the identities. $$ \c... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 40

Verify the identities. $$ \cot \left(\frac{A}{2}\right)-\tan \left(\frac{A}{2}\right)=2 \cot A $$

Short Answer

Expert verified
The identity is verified; both sides simplify to \( 2 \cot A \).

Step by step solution

01

Understand the Trigonometric Identities

We need to verify the identity: \( \cot \left(\frac{A}{2}\right)-\tan \left(\frac{A}{2}\right)=2 \cot A \). Recall that \( \cot\theta = \frac{\cos\theta}{\sin\theta} \) and \( \tan\theta = \frac{\sin\theta}{\cos\theta} \). Also, consider the identity \( \tan\left(\frac{A}{2}\right) = \frac{1 - \cos A}{\sin A} \) and \( \cot\left(\frac{A}{2}\right) = \frac{1 + \cos A}{\sin A} \). This information is the basis for starting our proof.
02

Express \( \cot \left(\frac{A}{2}\right) \) and \( \tan \left(\frac{A}{2}\right) \) in Terms of A

Using the half-angle identities for tangent and cotangent, we can express:- \( \cot\left(\frac{A}{2}\right) = \frac{1 + \cos A}{\sin A} \)- \( \tan\left(\frac{A}{2}\right) = \frac{1 - \cos A}{\sin A} \)Substitute these expressions into the left side of the given identity.
03

Simplify the Expression

Substitute the expressions from Step 2 into the identity:\[ \frac{1 + \cos A}{\sin A} - \frac{1 - \cos A}{\sin A} \]Combine the fractions:\[ \frac{(1 + \cos A) - (1 - \cos A)}{\sin A} = \frac{2\cos A}{\sin A} \].This simplifies to \( 2 \cot A \) since \( \cot A = \frac{\cos A}{\sin A} \).
04

Verify the Identity

We find that the expression \( \frac{2\cos A}{\sin A} \) indeed simplifies to \( 2 \cot A \), which matches the right side of the original identity. Therefore, the given 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.

Half-Angle Formulas
Half-angle formulas are essential in trigonometry. They allow us to express trigonometric functions of half-angles in terms of functions of the whole angle. These formulas are especially useful when simplifying expressions and working on proofs. For the angle \( \frac{A}{2} \), the half-angle formulas are given by:
  • \( \tan\left(\frac{A}{2}\right) = \frac{1 - \cos A}{\sin A} \)
  • \( \cot\left(\frac{A}{2}\right) = \frac{1 + \cos A}{\sin A} \)
These identities stem from the double angle relationships and are useful in breaking down complex trigonometric identities. By converting expressions involving \( \frac{A}{2} \) into functions of \( A \), we access a broader range of algebraic tools for simplification and proof.
Cotangent
Cotangent is one of the six fundamental trigonometric functions. It is the reciprocal of the tangent function. Mathematically, for an angle \( \theta \), it is expressed as:
  • \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \)
Cotangent is particularly helpful in converting expressions involving tangent into ratios of sine and cosine. In trigonometric identities like the one in this exercise, understanding cotangent's definition helps us manipulate and verify expressions. Using cotangent in conjunction with its half-angle identity simplifies the verification process.
Tangent
Tangent is another primary trigonometric function, defined for an angle \( \theta \) as:
  • \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
This function measures how an angle in a right triangle relates the opposite side to the adjacent side. In identity proofs, tangent can be rewritten using half-angle formulas to connect angles like \( \frac{A}{2} \) to the full angle \( A \), facilitating simplifications. It is crucial for converting trigonometric expressions, especially those involving sums or subtractions, into more manageable forms.
Proof Verification
Proof verification is a structured process of confirming the truth of a mathematical statement, in this case, a trigonometric identity. The primary goal is to show that both sides of an equation are equivalent.For this exercise, we started with the identity:\[ \cot\left(\frac{A}{2}\right) - \tan\left(\frac{A}{2}\right) = 2\cot A \]By substituting the half-angle formulas for cotangent and tangent, follow through:
  • Substitute: \( \cot\left(\frac{A}{2}\right) = \frac{1 + \cos A}{\sin A} \) and \( \tan\left(\frac{A}{2}\right) = \frac{1 - \cos A}{\sin A} \).
  • Combine fractions to simplify: \( \frac{(1 + \cos A) - (1 - \cos A)}{\sin A} \) becomes \( \frac{2\cos A}{\sin A} \).
  • Recognize: \( \frac{2\cos A}{\sin A} = 2\cot A \).
Completing these steps confirms the left-hand side equals the right-hand side, thus verifying the identity.

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

Verify the identities. $$ 2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)=\sin x $$

Use the half-angle identities to find the desired function values. If \(\csc x=3\) and \(\frac{\pi}{2}

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

An auger used to deliver grain to a storage bin can be raised and lowered, thus allowing for different size bins. Let \(\alpha\) be the angle formed by the auger and the ground for bin A such that \(\tan \alpha=\frac{24}{7}\). The angle formed by the auger and the ground for bin B is half of \(\alpha\). If the height \(h\), in feet, of a bin can be found using the formula \(h=75 \sin \theta\), where \(\theta\) is the angle formed by the ground and the auger, find the height of bin \(B\).

For Exercises 55 and 56, refer to the following: Monthly profits can be expressed as a function of sales, that is, \(p(s)\). A financial analysis of a company has determined that the sales \(s\) in thousands of dollars are also related to monthly profits \(p\) in thousands of dollars by the relationship: $$ \tan \theta=\frac{p}{s} \text { for } 0 \leq s \leq 50,0 \leq p<40 $$ Based on sales and profits, it can be determined that the domain for angle \(\theta\) is \(0 \leq \theta \leq 38^{\circ}\). If monthly profits are $$\$ 3000$$ and monthly sales are $$\$ 4000$$, find \(\tan \left(\frac{\theta}{2}\right)\).
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Decision 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.