Help showing $\sin(x)^{\sin(x)} < \cos(x)^{\cos(x)}$ between $x = 0$ and $x = \frac{\pi}{4}$? This is a homework problem that I just can't make progress on. I know that it can be shown through a special case of the generalized arithmetic-geometric mean inequality ($a^t b^{1-t} \leq ta + (1-t) b$), but I just can't choose variables correctly. Hoping for a hint, not a solution. I've tried setting $t = \sin^2 x$, $a = \sin x$, $b = \cos x$ (so $1-t = \cos^2 x$), and several permutations of these, but I'm getting nowhere.
 A: $t=\frac{\sin x}{\cos x}, 1-t=\frac{\cos x-\sin x}{\cos x}, a=\sin x, b=\cos x+\sin x$
Since $0<x<\frac{\pi}{4}$, we get $0<t<1, b=\frac{2}{\sqrt 2}\cos (\frac{\pi}{4}-x) > 1$
$a^t b^{1-t} \leq ta + (1-t) b$ translates to 
$(\sin x)^{\frac{\sin x}{\cos x}}b^{1-t} \le \cos x $.
Raising the above to power $\cos x$ and noting that $b>1, 1-t>0$, so $b^{\frac{1-t}{\cos x}}>1$, we get
$(\sin x)^{\sin x}<(\sin x)^{\sin x}b^{\frac{1-t}{\cos x}} \le (\cos x)^{\cos x}$, so done
A: Let $f(x)=x^{1/\sqrt{1-x^2}}$ for $x \in (0,1).$
$$\frac{f'(x)}{f(x)}=\frac{1-x^2+x^2 \ln x}{x (1-x^2)^{3/2}}$$
Now let $g(x)=1-x^2+x^2 \ln x$, $g'(x)= x(\ln x^2-1)<0,$ for $x \in (0,1].$
$g(x)$ is a decreasing function $g(x) >g(1)=0$, therefore $f'(x)>0$, $f(x$) is an increasing function. Finally,
$$ \sin x < \cos x,~ \mbox{for}~ x \in [0,\pi/4] \Rightarrow f(\sin x) < f(\cos x) \Rightarrow 
(\sin x) ^{\sec x} <(\cos x)^{\csc x}  \Rightarrow (\sin x)^{\sin x} < (\cos x)^{\cos x}.$$
A: When $x\in\left(0,\frac{\pi}{4}\right)$,
$$\sin x\in\left(0,\frac{1}{\sqrt{2}}\right),\ \tan x\in(0,1),\
\sin x+\cos x\in(1,\sqrt{2}).$$
It is easy to see that：
$$(\sin x)^{\sin x}<(\cos x)^{\cos x}
\iff(\sin x)^{^{\frac{\sin x}{\cos x}}}<\cos x.$$
Using Bernoulli's inequality$(1+x)^{\alpha}<1+\alpha x$ for $0<\alpha<1$ and $-1<x\neq0$,
\begin{align*}
(\sin x)^{^{\frac{\sin x}{\cos x}}}
  &=(1+(\sin x-1))^{^{\frac{\sin x}{\cos x}}}\\
  &<1+\frac{\sin x}{\cos x}(\sin x-1)\\
  &=\frac{\cos x+\sin^2 x-\sin x}{\cos x}.
\end{align*}
So we need only to prove that
$$\frac{\cos x+\sin^2 x-\sin x}{\cos x}<\cos x
\iff1<\cos x+\sin x.$$
