How do i find the exact value of tan⁻¹(tan ) ? i know the answer is negative, but i don't know why the inverse of tan and tan don't cancel out for the answer to just be 4pi/5 ?

This is because is a multivalued function and is not invertible over its entire domain. We restrict its domain to the interval , which gives one complete branch of values (or one period). and thus is outside the domain.

A different way to go about this is to find the value of first, then compute the inverse tangent of that result. But finding the trigonometric values of multiples of is somewhat tricky and perhaps more work than is needed.

Instead, we can use a trigonometric identity to find the value of whenever its argument falls outside the "standard" branch. 

We know that (because is -periodic), so . And now the function can be inverted, so that