Find the principal value of each of the following:
i) `cot^(-1)(-sqrt(3))`
ii) `sec^(-1)(-sqrt(2))`
iii) `"cosec"^(-1)(-1)`.

Explanation

i) We know that the range of the principal value of `cot^(-1)` is `(0,pi)`. Let `cot^(-)(-sqrt(3))=theta`. Then, `cottheta=-sqrt(3)=-cotpi/6=cot(pi-pi/6)=cot(5pi)/6`.
`therefore theta=(5pi)/6 in (0, pi)`.
Hence, the principal value of `cot^(-1)(-sqrt(3))` is `(5pi)/6`.
ii) We know that the range of principal value of `sec^(-1)` is `[0,pi]-{pi/2}`.
Let `sec^(-1)(-sqrt(2))=theta`. Then,
`sec^(-1)(-sqrt(2))=theta`. Then,
`sectheta=-sqrt(2)=-secpi/4 = sec(pi-pi/4)=sec(3pi)/4`.
`therefore theta=(3pi)/4 in [0,pi]-{pi/2}`.
Hence, the principal value of `sec^(-1)(-sqrt(2))` is `(3pi)/4`.
iii) We know that the range of the princiapl value of `"cosec"^(-1)` is `[-pi/2,pi/2]-{0}`.
Let `"cosec"^(-1)(-1)=theta`. Then, `"cosec"theta=-1`.
`"cosec"theta=-1=-"cosec"pi/2="cosec"(-pi/2)`.
`therefore theta=-pi/2 in [-pi/2, pi/2]-{0}`.
Hence, the principal value of `"cosec"^(-1)(-1)` is `pi/2`.