Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Join our platform to connect with experts ready to provide precise answers to your questions in different areas. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
To determine which expression is equivalent to [tex]\(\frac{1+\tan x}{-1+\tan x}\)[/tex], we need to recognize the trigonometric identities that simplify this expression. Let's systematically check each option to see which one matches our expression.
### Step-by-Step Solution:
1. Given Expression:
[tex]\[ \frac{1+\tan x}{-1+\tan x} \][/tex]
2. Option 1: [tex]\(\tan \left(\frac{3 \pi}{4} - x\right)\)[/tex]
We start by using the identity for the tangent of a difference of two angles:
[tex]\[ \tan \left(\frac{3 \pi}{4} - x\right) = \frac{\tan \left(\frac{3 \pi}{4}\right) - \tan x}{1 + \tan \left(\frac{3 \pi}{4}\right) \tan x} \][/tex]
Since [tex]\(\tan \left(\frac{3 \pi}{4}\right) = -1\)[/tex]:
[tex]\[ \tan \left(\frac{3 \pi}{4} - x\right) = \frac{-1 - \tan x}{1 + (-1) \cdot \tan x} = \frac{-1 - \tan x}{1 - \tan x} \][/tex]
To match the given expression, let's multiply both the numerator and denominator by [tex]\(-1\)[/tex]:
[tex]\[ \frac{-1 - \tan x}{1 - \tan x} = \frac{-(1 + \tan x)}{-(1 - \tan x)} = \frac{1 + \tan x}{-1 + \tan x} \][/tex]
This simplifies directly to the given expression. So, this option is correct.
3. Option 2: [tex]\(\tan \left(\frac{3 \pi}{4} + x\right)\)[/tex]
Using the addition formula for tangent:
[tex]\[ \tan \left(\frac{3 \pi}{4} + x\right) = \frac{\tan \left(\frac{3 \pi}{4}\right) + \tan x}{1 - \tan \left(\frac{3 \pi}{4}\right) \tan x} \][/tex]
Since [tex]\(\tan \left(\frac{3 \pi}{4}\right) = -1\)[/tex]:
[tex]\[ \tan \left(\frac{3 \pi}{4} + x\right) = \frac{-1 + \tan x}{1 - (-1) \cdot \tan x} = \frac{-1 + \tan x}{1 + \tan x} \][/tex]
This expression does not match the given expression. So, this option is incorrect.
4. Option 3: [tex]\(\tan \left(\frac{\pi}{4} - x\right)\)[/tex]
Using the identity for the tangent of a difference:
[tex]\[ \tan \left(\frac{\pi}{4} - x\right) = \frac{\tan \left(\frac{\pi}{4}\right) - \tan x}{1 + \tan \left(\frac{\pi}{4}\right) \tan x} \][/tex]
Since [tex]\(\tan \left(\frac{\pi}{4}\right) = 1\)[/tex]:
[tex]\[ \tan \left(\frac{\pi}{4} - x\right) = \frac{1 - \tan x}{1 + \tan x} \][/tex]
This does not match the given expression. So, this option is incorrect.
5. Option 4: [tex]\(\tan \left(\frac{\pi}{4} + x\right)\)[/tex]
Using the addition identity for tangent:
[tex]\[ \tan \left(\frac{\pi}{4} + x\right) = \frac{\tan \left(\frac{\pi}{4}\right) + \tan x}{1 - \tan \left(\frac{\pi}{4}\right) \tan x} \][/tex]
Since [tex]\(\tan \left(\frac{\pi}{4}\right) = 1\)[/tex]:
[tex]\[ \tan \left(\frac{\pi}{4} + x\right) = \frac{1 + \tan x}{1 - \tan x} \][/tex]
This expression does not match the given expression. So, this option is incorrect.
### Conclusion:
The given expression:
[tex]\[ \frac{1+\tan x}{-1+\tan x} \][/tex]
is equivalent to:
[tex]\[ \tan \left(\frac{3 \pi}{4} - x\right) \][/tex]
Thus, the correct option is:
[tex]\[ \tan \left(\frac{3 \pi}{4}-x\right) \][/tex]
### Step-by-Step Solution:
1. Given Expression:
[tex]\[ \frac{1+\tan x}{-1+\tan x} \][/tex]
2. Option 1: [tex]\(\tan \left(\frac{3 \pi}{4} - x\right)\)[/tex]
We start by using the identity for the tangent of a difference of two angles:
[tex]\[ \tan \left(\frac{3 \pi}{4} - x\right) = \frac{\tan \left(\frac{3 \pi}{4}\right) - \tan x}{1 + \tan \left(\frac{3 \pi}{4}\right) \tan x} \][/tex]
Since [tex]\(\tan \left(\frac{3 \pi}{4}\right) = -1\)[/tex]:
[tex]\[ \tan \left(\frac{3 \pi}{4} - x\right) = \frac{-1 - \tan x}{1 + (-1) \cdot \tan x} = \frac{-1 - \tan x}{1 - \tan x} \][/tex]
To match the given expression, let's multiply both the numerator and denominator by [tex]\(-1\)[/tex]:
[tex]\[ \frac{-1 - \tan x}{1 - \tan x} = \frac{-(1 + \tan x)}{-(1 - \tan x)} = \frac{1 + \tan x}{-1 + \tan x} \][/tex]
This simplifies directly to the given expression. So, this option is correct.
3. Option 2: [tex]\(\tan \left(\frac{3 \pi}{4} + x\right)\)[/tex]
Using the addition formula for tangent:
[tex]\[ \tan \left(\frac{3 \pi}{4} + x\right) = \frac{\tan \left(\frac{3 \pi}{4}\right) + \tan x}{1 - \tan \left(\frac{3 \pi}{4}\right) \tan x} \][/tex]
Since [tex]\(\tan \left(\frac{3 \pi}{4}\right) = -1\)[/tex]:
[tex]\[ \tan \left(\frac{3 \pi}{4} + x\right) = \frac{-1 + \tan x}{1 - (-1) \cdot \tan x} = \frac{-1 + \tan x}{1 + \tan x} \][/tex]
This expression does not match the given expression. So, this option is incorrect.
4. Option 3: [tex]\(\tan \left(\frac{\pi}{4} - x\right)\)[/tex]
Using the identity for the tangent of a difference:
[tex]\[ \tan \left(\frac{\pi}{4} - x\right) = \frac{\tan \left(\frac{\pi}{4}\right) - \tan x}{1 + \tan \left(\frac{\pi}{4}\right) \tan x} \][/tex]
Since [tex]\(\tan \left(\frac{\pi}{4}\right) = 1\)[/tex]:
[tex]\[ \tan \left(\frac{\pi}{4} - x\right) = \frac{1 - \tan x}{1 + \tan x} \][/tex]
This does not match the given expression. So, this option is incorrect.
5. Option 4: [tex]\(\tan \left(\frac{\pi}{4} + x\right)\)[/tex]
Using the addition identity for tangent:
[tex]\[ \tan \left(\frac{\pi}{4} + x\right) = \frac{\tan \left(\frac{\pi}{4}\right) + \tan x}{1 - \tan \left(\frac{\pi}{4}\right) \tan x} \][/tex]
Since [tex]\(\tan \left(\frac{\pi}{4}\right) = 1\)[/tex]:
[tex]\[ \tan \left(\frac{\pi}{4} + x\right) = \frac{1 + \tan x}{1 - \tan x} \][/tex]
This expression does not match the given expression. So, this option is incorrect.
### Conclusion:
The given expression:
[tex]\[ \frac{1+\tan x}{-1+\tan x} \][/tex]
is equivalent to:
[tex]\[ \tan \left(\frac{3 \pi}{4} - x\right) \][/tex]
Thus, the correct option is:
[tex]\[ \tan \left(\frac{3 \pi}{4}-x\right) \][/tex]
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
