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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 7: Q. 88 (page 487)

Write the trigonometric expression as an algebraic expression containing uand v
$\cos (\tan^{- 1} u + \tan^{- 1} v)$

Short Answer

Expert verified

$\cos (\tan^{- 1} u + \tan^{- 1} v) = (\sqrt{1 - u^{2}}) (\sqrt{1 - v^{2}}) (1 - u v)$

Step by step solution

01

Step 1. Given information is

$\cos (\tan^{- 1} u + \tan^{- 1} v)$

putting

$\tan^{- 1} u = x u = \tan x \tan^{- 1} v = y v = \tan y$

02

Step 2. Using trigonometry identity

$\frac{1}{\cos x} = \sqrt{1 + \tan^{2} x} \frac{1}{\cos x} = \sqrt{1 + u^{2}} \tan x = \frac{\sin x}{\cos x} \sin x = u \sqrt{1 + v^{2}}$

03

Step 3. Using the identity we get,

$\tan y = \frac{\sin y}{\cos y} v = \frac{\sin y}{\sqrt{1 + v^{2}}} \sin y = v \sqrt{1 + v^{2}}$

04

Step 4. Using the formulacosx+y=cosxcosy-sinxsiny

$= (\sqrt{1 + u^{2}}) (\sqrt{1 + v^{2}}) - (u \sqrt{1 + u^{2}}) (v \sqrt{1 + v^{2}}) = (\sqrt{1 + u^{2}}) (\sqrt{1 + v^{2}}) (1 - u v)$ On solving we get

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