# [Math] Show that, if $\tan^2(x) = 2\tan(x) +1,$ then $\tan(2x) = -1$

trigonometry

(i) Show that, if $\tan^2(x) = 2\tan(x) + 1,$ then $\tan (2x) = -1$

It's always the case that $\tan2x=2\tan x/(1-\tan^2x)$. If $\tan^2x =2\tan x+1$, then $2\tan x=\tan^2x-1$ and hence
$$\tan2x={2\tan x\over 1-\tan^2x}={\tan^2x-1\over1-\tan^2x}=-1$$