Looking for trustworthy answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Explore a wealth of knowledge from professionals across various disciplines on our comprehensive Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
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]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
