Find the Exact Value (No Calculator)

The tan inverse and the tangent just 'cancel' each other and you are left with  4pi/5....

   DO you see?   First you do the tangent.....then you do the tangent inverse...

        here is another example    sin^-1 ( sin 30)   =    sin^-1 ( 1/2 ) = 30

$\text{Consider the range of }\tan^{-1} x.\\ \tan^{-1} x \in \left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]\\ \text{That means we need to find a value } v\text{, which is between}-\dfrac{\pi}{2}\text{ and }\dfrac{\pi}{2}\text{, such that}\tan v = \tan\left(\dfrac{4\pi}{5}\right)\\ $

$\text{Solving the equation, }\\ v = \dfrac{4\pi}{5} + n\pi, \; n\in \mathbb Z$

Only when n = -1, v is inside the required range.

The solution we need is $v = \dfrac{4\pi}{5} - \pi = \dfrac{-\pi}{5}$.

Therefore $\tan^{-1}\left(\tan\left(\dfrac{4\pi}{5}\right)\right) = \dfrac{-\pi}{5}$