If `alpha in (-(3pi)/2,-pi)` , then the value of `tan^(-1)(cotalpha)-cot^(-1)(tanalpha)+sin^(-1)(sinalpha)+cos^(-1)(cosalpha)` is equal to

Options

A. A. `2 pi + alpha`
B. B. `pi + alpha`
C. C. `0`
D. D. `pi - alpha`

Answer

C

Explanation

For `alpha in (-(3pi)/(2), -pi), tan alpha lt 0`
`rArr tan^(-1) (cot alpha) - cot^(-1) (tan alpha)`
`= tan^(-1) (cot alpha) - [(pi)/(2) - tan^(-1) (tan alpha)]`
`= tan^(-1) (cot alpha) + tan^(-1) (tan alpha) - (pi)/(2) = - (pi)/(2) - (pi)/(2) = -pi`
Also for points in the second quadrant, we have
`sin^(-1) (sin alpha) + cos^(-1) (cos alpha) = pi`