Chapter 2: Q6P (page 71)

Verify each of the following by using equations (11.4), (12.2), and (12.3).
$\cos 2 z = c o s^{2} z - s i n^{2} z$

Short Answer

Expert verified

The equation $\cos 2 z = \cos^{2} z = \sin^{2} z$ is verified using the equations (11.4), (12.2) and (12.3).

Step by step solution

Given information

The given equation is, $\cos 2 z = \cos^{2} z - \sin^{2} z .$ .

Definition of Hyperbolic Function.

A hyperbolic function is a representation of the relationship between a point's distances from the origin to the coordinate axes as a function of an angle.

Use exponential form to expand the equation

The exponential form of the given equation is,

$\cos 2 z = \frac{e^{2 z i} + e^{- 2 z i}}{2}$ …. (1)

Multiply the equation (1) by $\frac{2}{2}$ .

$\cos 2 z = \frac{e^{2 z i} + e^{- 2 z i}}{2} \times \frac{2}{2} \cos 2 z = \frac{2 e^{2 z i} + 2 e^{- 2 z i}}{4}$

…. (2)

Add $2 [e^{z i} e^{- z i}] - 2 [e^{z i} e^{- z i}]$ to square the numerator.

$\cos 2 z = \frac{2 e^{2 z i} + 2 e^{- 2 z i} + 2 [e^{z i} e^{- z i}]}{4} + \frac{e^{2 z i} + e^{- 2 z i} + - 2 [e^{z i} e^{- z i}]}{4} \cos 2 z = {(\frac{e^{z i} + e^{- z i}}{2})}^{2} + {(\frac{e^{z i} + e^{- z i}}{2})}^{2} \cos 2 z = \cos^{2} z - \sin^{2} z$

Hence the equation 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.

Start your free trial

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

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Verify each of the following by using equations (11.4), (12.2), and (12.3).
$\cos 2 z = c o s^{2} z - s i n^{2} z$

Short Answer

Step by step solution

Given information

Definition of Hyperbolic Function.

Use exponential form to expand the equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electromagnetism

Astrophysics

Fields in Physics

Fundamentals of Physics

Linear Momentum

Wave Optics

Study anywhere. Anytime. Across all devices.

Verify each of the following by using equations (11.4), (12.2), and (12.3).cos2z=cos2z-sin2z

Short Answer

Step by step solution

Given information

Definition of Hyperbolic Function.

Use exponential form to expand the equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electromagnetism

Astrophysics

Fields in Physics

Fundamentals of Physics

Linear Momentum

Wave Optics

Study anywhere. Anytime. Across all devices.

Verify each of the following by using equations (11.4), (12.2), and (12.3).
$\cos 2 z = c o s^{2} z - s i n^{2} z$