Which of the following is/are correct? `(tan"x")^(I n(cosx))<(cot"x")^(in(cosx))AAx in (pi/4,pi/2)` `(sin"x")^(I n(secx))>(cos"x")^(in(cosx))AAx in (0,pi/4)` `(s e cpi/3)^(I n(tanx))` >`(s e cpi/3)^(I n(cosx))` `AAx in (pi/4,pi/2)` `(1/2)^(I n(sinx)>(3/4)^(I n(sinx)AAx 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.