Question

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)

Answer 1

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.

Answer 2

`\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 \;$\boxed{\tan \left((3 \pi)/(4)+x\right)}$\:.}`