IDNLearn.com: Your trusted source for accurate and reliable answers. Join our community to receive timely and reliable responses to your questions from knowledgeable professionals.
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]
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. Thank you for trusting IDNLearn.com with your questions. Visit us again for clear, concise, and accurate answers.
