A question for professors or college students.?

Question

Show this inequality:

(cos x)^(cos x)>(sin x)^(sin x)

0

Answer 1

when x = pi/4
then cos x = sin x = 1 / sq rt(2)
so, (cos x)^(cos x) = (sin x)^(sin x)

but in other cases the value of cos x is not equals to sin x

so within the range [ 0 < x < pi/4 ], (cos x)^(cos x) is not equals to (sin x)^(sin x)

now within this range always cos x > sin x [ if u take any value within this range and evaluate the sine value & cosine value of it then u will find this relation ]

so, (cos x)^(cos x) > (sin x)^(sin x) [proved]

Answer 2

cos² x - sin² x = cos 2x,
which is positive on [0, π/4).

Answer 3

for 0
cosx > sinx ...........................1
so
log cosx > sinx .....................2

from equations 1 & 2

cosx log(cosx) > sinx log(sinx)

taking antilog of above equation

(cosx)^(cosx) > (sinx)^(sinx)
hence solved.