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The expression -1+tanx/1+tanx can be rewritten as which of the following?

A. tan((3pi/4) - x)
B. tan((5pi/4) - x)
C. tan((3pi/4) + x)
D. tan((5pi/4) + x)

User Thamina
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2 Answers

4 votes

To rewrite the expression -1 + tan(x) / (1 + tan(x)), we can start by finding a common denominator for the terms in the numerator.

Multiplying the numerator and denominator by 1 - tan(x), we get:

((-1 + tan(x)) * (1 - tan(x))) / ((1 + tan(x)) * (1 - tan(x)))

Expanding the numerator, we have:

(-1 + tan(x) - tan(x) + tan^2(x)) / (1 - tan^2(x))

Simplifying, we get:

(tan^2(x) - 2tan(x) - 1) / (-tan^2(x) + 1)

Now, we can factor the numerator and denominator:

[(tan(x) - 1)(tan(x) + 1)] / [(1 - tan(x))(1 + tan(x))]

Canceling out the common factors, we have:

-(tan(x) - 1) / (1 - tan(x))

Rearranging, we get:

(1 - tan(x)) / (tan(x) - 1)

Therefore, the expression -1 + tan(x) / (1 + tan(x)) can be rewritten as:

(1 - tan(x)) / (tan(x) - 1)

Comparing this expression to the answer choices given, none of the options A, B, C, or D matches the rewritten expression.

User Corrado Piola
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8.3k points
4 votes

Answer:


\textsf{C.} \quad \tan \left((3\pi)/(4)+x\right)

Explanation:

Given trigonometric expression:


(-1+ \tan x)/(1+\tan x)

Rewrite the expression as:


((-1)+\tan x)/(1-(-1)\tan x)

This is now in the form of the tangent trigonometric identity:


\boxed{\begin{minipage}{5cm}\underline{Trigonometric Identity}\\\\$\tan(A+B)=(\tan A + \tan B)/(1 - \tan A \tan B)$\\\end{minipage}}

Using this identity, let tan A = -1 and tan B = tan x:


\tan(A+x)=((-1)+\tan x)/(1-(-1)\tan x)


\textsf{As the solution of\;\;$\tan A=-1$\;\;is\;\;$A=(3\pi)/(4)+\pi n$,\;\;then:}


\tan \left((3 \pi)/(4)+x\right)=(\tan \left((3 \pi)/(4)\right)+\tan x)/(1-\tan \left((3 \pi)/(4)\right)\tan x)


\tan \left((3 \pi)/(4)+x\right)=((-1)+\tan x)/(1-(-1)\tan x)


\tan \left((3 \pi)/(4)+x\right)=(-1+\tan x)/(1+\tan x)


\textsf{Therefore,\;\;$(-1+\tan x)/(1+\tan x)$\;\;can be rewritten as \;$\boxed{\tan \left((3 \pi)/(4)+x\right)}$\:.}

User HappyFace
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7.9k points