Short Answer

Establish each identity.
$\frac{1 + \cos θ + \sin θ}{1 + \cos θ - \sin θ} = s e c θ + \tan θ$

The left side of the given equation has been simplified to bring the right side.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information

The given equation is $\frac{1 + \cos θ + \sin θ}{1 + \cos θ - \sin θ} = s e c θ + \tan θ$

Step 2. Calculation

Consider the left side of the given equation and simplify to bring the right side in order to establish the equation.

$\frac{1 + \cos θ + \sin θ}{1 + \cos θ - \sin θ} = \frac{\frac{1 + \cos θ + \sin θ}{\cos θ}}{\frac{1 + \cos θ - \sin θ}{\cos θ}} = \frac{\frac{1}{\cos θ} + \frac{\cos θ}{\cos θ} + \frac{\sin θ}{\cos θ}}{\frac{1}{\cos θ} + \frac{\cos θ}{\cos θ} - \frac{\sin θ}{\cos θ}} = \frac{s e c θ + 1 - \tan θ}{s e c θ + 1 - \tan θ} = \frac{\tan θ + s e c θ - (s e c^{2} θ - \tan^{2} θ)}{\tan θ - s e c θ + 1} = \frac{\tan θ + s e c θ - (s e c θ - \tan θ) (s e c θ + \tan θ)}{\tan θ - s e c θ + 1} = \frac{(\tan θ + s e c θ) (1 - (s e c θ + \tan θ))}{\tan θ - s e c θ + 1} = \tan θ + s e c θ$

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Precalculus Enhanced with Graphing Utilities

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Short Answer

Establish each identity.
$\frac{1 + \cos θ + \sin θ}{1 + \cos θ - \sin θ} = s e c θ + \tan θ$

Step by Step Solution

Step 1. Given Information

Step 2. Calculation

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Precalculus Enhanced with Graphing Utilities

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Short Answer

Establish each identity. 1+cosθ+sinθ1+cosθ-sinθ=secθ+tanθ

Step by Step Solution

Step 1. Given Information

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Establish each identity.
$\frac{1 + \cos θ + \sin θ}{1 + \cos θ - \sin θ} = s e c θ + \tan θ$