To determine To establish: The identity cos ( α + β ) cos ( α − β ) = 1 − tan α ⋅ tan β 1 + tan α ⋅ tan β . Answer Solution: The identity cos ( α + β ) cos ( α − β ) = 1 − tan α ⋅ tan β 1 + tan α ⋅ tan β is established. Explanation Given: cos ( α + β ) cos ( α − β ) = 1 − tan α ⋅ tan β 1 + tan α ⋅ tan β Formula used: By sum and difference formula of cosine . cos ( α + β ) cos ( α − β ) = cos α cos β − sin α sin β cos α cos β + sin α sin β Calculation: Left hand side, cos ( α + β ) cos ( α − β ) Dividing by cos α cos β . cos ( α + β ) cos ( α − β ) = 1 − tan α ⋅ tan β 1 + tan α tan β LHS = RHS

9th Edition

ISBN: 9780321716835

Author: Michael Sullivan

Publisher: Addison Wesley

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Textbook Question

Chapter 7.5, Problem 64AYU

establish each identify.

$\frac{\cos (α + β)}{\cos (α - β)} = \frac{1 - \tan α tan β}{1 + \tan α tan β}$

Expert Solution & Answer

To determine

To establish: The identity $\frac{cos (α + β)}{\cos (α - β)} = \frac{1 - \tan α \cdot \tan β}{1 + \tan α \cdot \tan β}$ .

Answer to Problem 64AYU

Solution:

The identity $\frac{cos (α + β)}{\cos (α - β)} = \frac{1 - \tan α \cdot \tan β}{1 + \tan α \cdot \tan β}$ is established.

Explanation of Solution

Given:

$\frac{cos (α + β)}{\cos (α - β)} = \frac{1 - \tan α \cdot \tan β}{1 + \tan α \cdot \tan β}$

Formula used:

By sum and difference formula of $cosine$ .

$\frac{\cos (α + β)}{\cos (α - β)} = \frac{\cos α \cos β - \sin α \sin β}{\cos α \cos β + \sin α \sin β}$

Calculation:

Left hand side,

$\frac{\cos (α + β)}{\cos (α - β)}$

Dividing by $\cos α \cos β$ .

$\frac{\cos (α + β)}{\cos (α - β)} = \frac{1 - \tan α \cdot \tan β}{1 + \tan α \tan β}$

$LHS = RHS$

Chapter 7.5, Problem 63AYU

Chapter 7.5, Problem 65AYU

Chapter 7 Solutions

Precalculus

Ch. 7.1 - Prob. 1AYU Ch. 7.1 - Prob. 2AYU Ch. 7.1 - Prob. 3AYU Ch. 7.1 - Prob. 4AYU Ch. 7.1 - Prob. 5AYU Ch. 7.1 - Prob. 6AYU Ch. 7.1 - y= sin 1 x means _____, where 1x1 and 2 y 2 .Ch. 7.1 - Prob. 8AYU Ch. 7.1 - Prob. 9AYU Ch. 7.1 - Prob. 10AYU

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establish each identify. cos ( α + β ) cos ( α − β ) = 1 − tan α tan β 1 + tan α tan β

Solution Summary: The author explains that the identity cos = 1 tan, 1 + a corresponding sum and difference formula.