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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. 60 (page 496)

In Problems 47鈥� 68, establish each identity.
$\tan \frac{v}{2} = c s c v - c o t v$ .

Short Answer

Expert verified

The identity $\tan \frac{v}{2} = c s c v - c o t v$ is established.

Step by step solution

01

Step 1 Given identity is,

$\tan \frac{v}{2} = c s c v - c o t v$ .

The formulae used in this problem are,

$\sin 胃 = 2 \sin \frac{胃}{2} \cos \frac{胃}{2} \cos 胃 = 1 - 2 \sin^{2} \frac{胃}{2} \sin (2 胃) = 2 \sin 胃 \cos 胃 \cos (2 胃) = 1 - 2 \sin^{2} 胃$

02

Step 2 Use the formula in the required places and prove that the value of expressions in the both sides are equal.

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

Hence, proved.

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Most popular questions from this chapter

In Problems 9鈥�36, find the exact value of each expression.

$\cos^{- 1} (\sin \frac{5 蟺}{4})$

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

Express the product $\cos (4 胃) \cos (2 胃)$ as a sum containing only sines or only cosines.

True or False:

$\tan 2 胃 + \tan 2 胃 = \tan (4 胃)$

solve each equation on the interval

$0 鈮� 胃 < 2 蟺$

$\sin 胃 + \cos 胃 = \sqrt{2}$

See all solutions

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