Find the principal value of `tan^(-1)sqrt(3)-sec^(-1)(-2)`

Explanation

We know that the ranges of principal values of `tan^(-1)` and `sec^(-1)` are `(-pi/2, pi/2)` and `[0,pi]-{pi/2} rArr theta_(1)=pi/3 in (-pi/2,pi/2)`.
Let `sec^(-1)(-2)=theta_(2)`. Then,
`tantheta_(1)=sqrt(3)=tanpi/3 rArr theta_(1)=pi/3 in (-pi/2, pi/2)`.
Let `sec^(-1)(-2)=theta_(2)`. Then,
`sectheta_(2)=-2=-secpi/3=sec(pi-pi/3)=sec(2pi)/3`.
`therefore theta_(2)=(2pi)/3 in [0,pi]-{pi/2}`.
Hence, `tan^(-1)sqrt(3)-sec^(-1)(-2)=(pi/3-(2pi)/3)=-pi/3`.