If $\tan x = -1$, simplify $\tan(\pi/3+x)$

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If $\tan x = -1$, simplify $\tan(\pi/3+x)$.

Progress so far:

See new expression in top right. What do I do to rationalize this fraction's denominator? Do I multiply the fraction by 1+sqrt3/1+sqrt3?

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EDIT:

Okay, I'm a little confused as to what my 'final answer' will be.. mfl's answer seems as if it should be simplified. If so, which answer below is correct? Or is mfl's answer to be left as is?

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2 Answers

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You are on the right way, but you have forgotten to substitute $\tan x=1$ in all places where it appears.

$$\tan(\frac{\pi}{3}+x)=\frac{\tan\frac{\pi}{3}+\tan x}{1-\tan\frac{\pi}{3}\tan x}\underbrace{=}_{\tan x=-1}\frac{\sqrt{3}-1}{1+\sqrt{3}}\\=\frac{\sqrt{3}-1}{1+\sqrt{3}}\frac{\sqrt{3}-1}{\sqrt{3}-1}=\frac{(\sqrt{3}-1)^2}{3-1}=\frac{(\sqrt{3}-1)^2}{2}.$$

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Since $\tan x=-1$, your denominator should be $1+\sqrt{3}$.

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