Which of the following is correct ?

Options

A. `(tan^(-1)x-tan^(-1)y)/(x-y)le1 AAx,y in R,(xney)`
B. `(sin^(-1)x-sin^(-1)y)/(x-y)gt1 AAx, y in [-1,1], x ney`
C. `(cos^(-1)x-cos^(-1)y)/(x-y)lt1 AAx, y in [-1,1], x ney`
D. `(cot^(-1)x-cot^(-1)y)/(x-y)lt1 AAx, y in R, x ney`

Answer

A, B

Explanation

All four fucntion `tan^(-1)x, sin^(-1)x, cos^(-1)x, cot^(-1)x` are continuous and differentiable in their domain.
So by L. M. V. T.
(a) `(tan^(-1)x-tan^(-1)y)/(x-y)=(1)/(1+e^(2))le1`
(b) `(sin^(-1)x-sin^(-1)y)/(x-y)=(1)/(sqrt(1-e^(2)))le1`
(c) `(cos^(-1)x-cos^(-1)y)/(y-x)=(1)/(sqrt(1-e^(2)))le1`
(d) `(cot^(-1)x-cot^(-1)x)/(y-x)=(1)/(1+e^(2))le1`