Which of the following is/are correct ?

Options

A. `(tanx)^("In"(sinx))gt(cotx)^("In"(sinx)), AA x in(0,pi//4)`
B. `4^("In"cosecx)lt5^("In"cosecx), AA x in(0,pi//2)`
C. `(1//2)^("In"(cosx))lt(1//3)^("In"(cosx)), AA x in(0,pi//2)`
D. `2^("In"(tanx))lt2^("In"(tanx)), AA x in(0,pi//2)`

Answer

A, B, C, D

Explanation

(1) For ` x in (0,pi/4), tanxltcotx`
Also `In(sinx)lt0`
`rArr (tanx)^(in(sinx))gt(cotx)^(in(sinx))`
(2) For `x in (0,pi/2),cosecxge1`
`rArr In(cosecx)ge0`
`rArr 4^(In(cosecx))lt5^(in(cosecx))`
`(3) x in (0,pi/2)`
`rArr cosx in (0,1)`
`rArr In(cosx)lt0`
Also `1/2gt1/3`
`rArr (1/2)^(In(cosx))lt(1/3)^(In(cosx))`
(4) For `x in (0,pi/2)`
Since `sinxlttanx`,we get `In(sinx)ltIn(tanx)`
`rArr2^(In(sinx))lt2^(In(tanx))`