# How do you find the exact value of tan^-1(tan((2pi)/3))?

These are inverses of each other. ${\tan}^{- 1} \left(x\right)$ (or $\arctan x$) is the inverse of $\tan \left(x\right)$.
Let $A \left(x\right)$ be a function, and let ${A}^{- 1} \left(x\right)$ be its inverse (note that this is not the same as the reciprocal).
Then, the function composition of ${A}^{- 1} \left(x\right)$ with $A \left(x\right)$ is ${A}^{- 1} \left(A \left(x\right)\right) = x$.
Since the domain of $\tan x$ is $\left(\frac{- \pi}{2} , \frac{\pi}{2}\right) \pm \pi k$ (where $k$ is in the set of integers), and the period is $\pi$, take the coterminal angle to be $- \frac{\pi}{3}$.
So the exact answer is $- \frac{\pi}{3}$.