MHB Proving an Inequality: $(\sin\, x)^{\sin\, x}\,<(\cos\, x)^{\cos\, x}$

[kaliprasad]

One of the 2 inequalities

$(\sin\, x)^{\sin\, x}\,<(\cos\, x)^{\cos\, x} $ and $(\sin\, x)^{\sin\, x}\,>(\cos\, x)^{\cos\, x} $ is always true for all x such that $0 \,< \, x \, < \pi/4$ Identify the inequality and prove it

 

[Albert1]

One of the 2 inequalities

$(\sin\, x)^{\sin\, x}\,<(\cos\, x)^{\cos\, x} $ and $(\sin\, x)^{\sin\, x}\,>(\cos\, x)^{\cos\, x} $ is always true for all x such that $0 \,< \, x \, < \pi/4$ Identify the inequality and prove it

Spoiler

$(\sin\, x)^{\sin\, x}\,<(\cos\, x)^{\cos\, x} $
for :$0\,<\,x<\,\pi/4$
$sin\, x < cos \,x$
if $a<b$ then
$a^a<b^b$
(here :$a>0 ,\,and \,\, b>0)$