The value of tan^-1 (√3) - sec^-1(-2) is equal to-

-π/3

Let  tan^-1 (√3) = x. Then, tan x =√3 = tan (π/3) 

We know that the range of principal value branch of tan^-1  is (-π/2 , π/2)

so tan^-1 (√3) = (π/3)

Let, sec^-1(-2) = y. Then secy = 2 = -sec(π/3) = sec(π- π/3) = sec(2π/3)

We know that the range of principal value branch of sec^-1  is [0, π]- {π/2}

so sec^-1(-2) = 2π/3

Hence,  tan^-1 (√3) - sec^-1(-2) = (π/3)- (2π/3) = -π/3