## Trigonometry (11th Edition) Clone

$$\tan(2\pi-x)=-\tan x$$
$$X=\tan(2\pi-x)$$ According to tangent difference identity: $$\tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}$$ Expand $X$: $$X=\frac{\tan2\pi-\tan x}{1+\tan2\pi\tan x}$$ In the trigonometric circle, point $2\pi$ collides with point $0$. Therefore, $\tan2\pi=\tan0=0$. $$X=\frac{0-\tan x}{1+0\times\tan x}$$ $$X=\frac{-\tan x}{1}$$ $$X=-\tan x$$ Overall, $$\tan(2\pi-x)=-\tan x$$