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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. 100 (page 487)

Show that $c o t^{- 1} e^{v} = \tan^{- 1} e^{- v}$

Short Answer

Expert verified

The equation $c o t^{- 1} e^{v} = \tan^{- 1} e^{- v}$ can be obtained as

$c o t^{- 1} e^{v} = A 鈰� (i) e^{v} = c o t A e^{- v} = \tan A \tan^{- 1} e^{v} = A 鈰� (i i)$

from $i & i i$

$c o t^{- 1} e^{v} = \tan^{- 1} e^{- v}$

Step by step solution

01

Step 1. Given data

The given equation for verification is

$c o t^{- 1} e^{v} = \tan^{- 1} e^{- v}$

02

Step 2. Proof

Consider

$c o t^{- 1} e^{v} = A 鈰� (1)$

Change inverse trigonometric into normal trigonometric function

$e^{v} = c o t A$

and $\frac{1}{e^{v}} = \frac{1}{c o t A} e^{- v} = \tan A \tan^{- 1} (e^{- v}) = A 鈰� (2)$

From Equation $1 & 2$

$c o t^{- 1} e^{v} = \tan^{- 1} e^{- v}$

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