Question

​Find the principal values of the following:

`cos^-1(tan  (3pi)/4)`

Answer 1

Let `cos^-1(tan  (3pi)/4)=y`

Then, 

`cosy=tan  (3pi)/4`

We know that the range of the principal value branch is [0,pi]

thus, 

`cosy=tan  (3pi)/4=-1=cos(pi)`

`=>y=piin[0,pi]`

Hence, the principal value of `cos^-1(tan  (3pi)/4)` is  π.