Chapter 7: Q. 40 (page 504)

In Problem $28 - 44$ , Establish each identity
$2 c o t θ c o t (2 θ) = c o t^{2} θ - 1$

Short Answer

Expert verified

The derivation of the equation $2 c o t θ c o t (2 θ) = c o t^{2} θ - 1$ is

role="math" localid="1646510729288" $2 c o t θ c o t (2 θ) = 2 \frac{\cos θ \cos (2 θ)}{\sin θ \sin (2 θ)} = 2 \frac{\cos θ (\cos^{2} θ - \sin^{2} θ)}{\sin θ (2 \sin θ \cos θ)} = \frac{\cos^{2} θ - \sin^{2} θ}{\sin^{2} θ} = c o t^{2} θ - 1$

Step by step solution

Step 1. Given data

The given equation for derivation is

$2 c o t θ c o t (2 θ) = c o t^{2} θ - 1$

Step 2. Use double angle formula

Write the left-hand side expression in terms of sine and cosine functions

$2 c o t θ c o t (2 θ) = 2 \frac{\cos θ \cos (2 θ)}{\sin θ \sin (2 θ)}$

Use double angle formula

$2 c o t θ c o t (2 θ) = 2 \frac{\cos θ \cos (2 θ)}{\sin θ \sin (2 θ)} = 2 \frac{\cos θ (\cos^{2} θ - \sin^{2} θ)}{\sin θ (2 \sin θ \cos θ)} = \frac{\cos^{2} θ - \sin^{2} θ}{\sin^{2} θ}$

Step 3. Proof

Rearrange the equation

$2 c o t θ c o t (2 θ) = \frac{\cos^{2} θ - \sin^{2} θ}{\sin^{2} θ} = \frac{\cos^{2} θ}{\sin^{2} θ} - \frac{\sin^{2} θ}{\sin^{2} θ} = c o t^{2} θ - 1$

The left-hand side expression is equal to the right-hand side expression

So identity is stabilised

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Problem $28 - 44$ , Establish each identity
$2 c o t θ c o t (2 θ) = c o t^{2} θ - 1$

Short Answer

Step by step solution

Step 1. Given data

Step 2. Use double angle formula

Step 3. Proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Discrete Mathematics

Pure Maths

Logic and Functions

Mechanics Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Problem28-44, Establish each identity2cotθcot2θ=cot2θ-1

Short Answer

Step by step solution

Step 1. Given data

Step 2. Use double angle formula

Step 3. Proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Discrete Mathematics

Pure Maths

Logic and Functions

Mechanics Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Problem $28 - 44$ , Establish each identity
$2 c o t θ c o t (2 θ) = c o t^{2} θ - 1$