Indicate the relation which can hold in their respective domain for infinite values of `x: ` `(a) tan"|tan^(-1)x|=|x|` (b) `"cot"|cot^(-1)x|=|x|` `(c) tan^(-1)|tanx|=|x|` (d) `sin|sin^(-1)x|=|x|`

Options

A. `tan|tan^(-1) x| = |x|`
B. `cot |cot^(-1) x| = |x|`
C. `tan^(-1) |tan x| = |x|`
D. `sin |sin^(-1) x| = |x|`

Answer

A, B, C, D

Explanation

Since `|tan^(-1)x| = {(tan^(-1) x," if " x ge 0),(-tan^(-1) x," if " x lt 0):}`
`rArr |tan^(-1)x| = tan^(-1) |x| AA x in R`
`rArr tan |tan^(-1) x| = tan tan^(-1) |x| = |x|`
Also, `|cot^(-1) x| = cot^(-1) x, AA x in R`
`rArr cot|cot^(-1) x| = x, AA x in R`
`tan^(-1)|tan x| = {(x,"if " 0 lt x lt (pi)/(2)),(-x," if " -(pi)/(2) lt x lt 0):}`
`sin|sin^(-1)x| = {(x,x in [0, 1]),(-x,x in [-1, 0]):}`