Question

The expression the (-1 + tan x)/(1 + tan x) can be rewritten as which of the following?tan((3pi/4) - x)tan((5pi/4) - x)tan((3pi/4) + x)tan((5pi/4) + x)

Answer 1

To answer this question we need to remember the identity:

$\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$

and that

$\tan\frac{3\pi}{4}=-1$

If we let A be 3π/4 and B be x, we have:

$\begin{gathered} \tan(\frac{3\pi}{4}+x)=\frac{\tan\frac{3\pi}{4}+\tan x}{1-\tan\frac{3\pi}{4}\tan x} \\ =\frac{-1+\tan x}{1-(-1)\tan x} \\ =\frac{-1+\tan x}{1+\tan x} \end{gathered}$

Therefore, we have that:

$\frac{-1+\tan x}{1+\tan x}=\tan(\frac{3\pi}{4}+x)$