Answer :
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]
