Which of the following is/are correct ? (a) `(tanx)^(In (cosx))lt(cotx)^(In(cosx))AAx in(pi/4,pi/2 )` (b)`(sinx)^(In (secx))gt(cosx)^(In(secx))AA x in(0,pi/4)` (c ) `(sec. pi/3)^(In (tanx))gt(sec. pi/3)^(In(cosx))AA x in (pi/4,pi/2)` (d) `(1/2)^(In(sinx))gt(3/4)^(In(sinx))AA x in(0,pi/2)`

Explanation

(a) For `pi/4lt xlt pi/2,tanxgtcotx`
but In `(cosx)lt0`
`:. (tanx)^(In(secx))lt(cosx)^(In(secx))`
Hence, the statement is correct.
(b) For `x in (0,pi/4),cosxgt sinx`
but In (secx) gt 0 (as sec x gt1)
`:. (sinx)^(In(secx))lt(cosx)^(In(sexx))`
Hence, the statement is not correct
(c ) For `x in (pi/4,pi/2),tanxgt1`
`:. In (tanx)gt0`
but In`(cosx)lt0`
`:. (sec. pi/3)^(In(tanx))gt (sec. pi/3)^(In(cosx))`
Hence, the statement is correct.
(d) For `x in(0,pi/2)`
In `(sinx)lt0`
as `1/2lt3/4`
`:. (1/2)^(In(sinx))gt(3/4)^(In(sinx))`
Hence, the statement is correct.