Q. 37 Establish each identity. 聽sin伪... [FREE SOLUTION]

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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. 37 (page 501)

Establish each identity.
$\frac{\sin 伪 + \sin 尾}{\sin 伪 - \sin 尾} = - \tan \frac{伪 + 尾}{2} \cot \frac{伪 - 尾}{2}$

Short Answer

Expert verified

The equation $\frac{\sin 伪 + \sin 尾}{\sin 伪 - \sin 尾} = \tan \frac{伪 + 尾}{2} \cot \frac{伪 - 尾}{2}$ is established as an identity.

Step by step solution

Step 1. Given Information

We are given the equation $\frac{\sin 伪 + \sin 尾}{\sin 伪 - \sin 尾} = \tan \frac{伪 + 尾}{2} \cot \frac{伪 - 尾}{2}$ and we need to establish it as an identity.

Step 2. Evaluating the L.H.S.

We first simplify the L.H.S. using the sum to product formula.

The standard formulae required are

$\sin 伪 + \sin 尾 = 2 \sin (\frac{伪 + 尾}{2}) \cos (\frac{伪 - 尾}{2}) \sin 伪 - \sin 尾 = 2 \sin (\frac{伪 - 尾}{2}) \cos (\frac{伪 + 尾}{2})$

Using the above formulae we get,

$\frac{\sin 伪 + \sin 尾}{\sin 伪 - \sin 尾} = \frac{2 \sin (\frac{伪 + 尾}{2}) \cos (\frac{伪 - 尾}{2})}{2 \sin (\frac{伪 - 尾}{2}) \cos (\frac{伪 + 尾}{2})} = \frac{\sin (\frac{伪 + 尾}{2})}{\cos (\frac{伪 + 尾}{2})} 脳 \frac{\cos (\frac{伪 - 尾}{2})}{\sin (\frac{伪 - 尾}{2})}$

Using the identities

role="math" localid="1646483001088" $\tan 胃 = \frac{\sin 胃}{\cos 胃} 肠辞迟胃 = \frac{\cos 胃}{\sin 胃}$

we get,

$\frac{\sin (\frac{伪 + 尾}{2})}{\cos (\frac{伪 + 尾}{2})} 脳 \frac{\cos (\frac{伪 - 尾}{2})}{\sin (\frac{伪 - 尾}{2})} = \tan (\frac{伪 + 尾}{2}) \cot (\frac{伪 - 尾}{2})$

Thus, we can see that the L.H.S. of the identity is equal to the R.H.S. of the identity.

Therefore, the given equation is established as an identity.

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91影视

Establish each identity.
$\frac{\sin 伪 + \sin 尾}{\sin 伪 - \sin 尾} = - \tan \frac{伪 + 尾}{2} \cot \frac{伪 - 尾}{2}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Evaluating the L.H.S.

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91影视

Establish each identity. sin伪+sin尾sin伪-sin尾=-tan伪+尾2cot伪-尾2

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Evaluating the L.H.S.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

Calculus

Geometry

Probability and Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Establish each identity.
$\frac{\sin 伪 + \sin 尾}{\sin 伪 - \sin 尾} = - \tan \frac{伪 + 尾}{2} \cot \frac{伪 - 尾}{2}$