Proving $(\sin^2 \alpha+\sin\alpha \cos \alpha)^{\sin \alpha}(\cos^2 \alpha+\sin \alpha \cos \alpha)^{\cos \alpha}\leq 1$ 
If $\alpha \in \left(0, \frac{\pi}{2}\right)$, prove that:
$$(\sin^2 \alpha+\sin\alpha \cos \alpha)^{\sin \alpha}(\cos^2 \alpha+\sin \alpha \cos \alpha)^{\cos \alpha}\leq 1$$

I know $\sin \alpha$ and $\cos \alpha$ are positive over $\alpha \in \left(0, \frac{\pi}{2}\right)$, so with $a=\sin \alpha,\ b=\cos \alpha$, $a>0,\ b>0$ and $a^2+b^2=1$, the inequality is
$$(a^2+ab)^a(b^2+ab)^b \leq 1$$
$$\Leftarrow a^ab^b(a+b)^{a+b} \leq 1$$
and here I don't know how to prove this inequality.
 A: Rewrite the last line of your proof as: $a^{\frac{a}{a+b}}\cdot b^{\frac{b}{a+b}}\cdot (a+b) \le 1$. Apply the weighted AM-GM inequality you have: $LHS \le (\dfrac{a^2}{a+b}+\dfrac{b^2}{a+b})\cdot (a+b) = \dfrac{1}{a+b}\cdot (a+b) = 1 = RHS$. 
A: Passing to logarithm, we can write the inequality as:
$$\sin \alpha \cdot \ln (\sin \alpha) + \cos \alpha \cdot \ln(\cos \alpha)+ (\sin \alpha+\cos \alpha)\cdot \ln(\sin \alpha+\cos \alpha)\leq 0$$
or
$$\frac{\sin \alpha}{\sin \alpha+\cos \alpha}\cdot \ln(\sin \alpha)+\frac{\cos \alpha}{\sin \alpha+\cos \alpha}\cdot \ln(\cos \alpha) \leq -\ln (\sin \alpha+\cos \alpha)$$
or
$$\frac{\sin \alpha}{\sin \alpha+\cos \alpha}\cdot \ln(\sin \alpha)+\frac{\cos \alpha}{\sin \alpha+\cos \alpha}\cdot \ln(\cos \alpha)\\ \leq \ln \left[\frac{\sin \alpha}{\sin \alpha+\cos \alpha}\cdot \sin \alpha+\frac{\cos \alpha}{\sin \alpha+\cos \alpha}\cdot \cos \alpha\right]$$
This is a direct application of Jensen's inequality because $\ln$ is concave on $(0, \infty)$.
A: The inequality is equivalent to: $$(1-\cos^2 \alpha+\sin\alpha \cos \alpha)^{\sin \alpha}(1-\sin^2 \alpha+\sin \alpha \cos \alpha)^{\cos \alpha}\leq 1$$
By Bernoulli Inequality, $$(1-\cos^2 \alpha+\sin\alpha \cos \alpha)^{\sin \alpha}(1-\sin^2 \alpha+\sin \alpha \cos \alpha)^{\cos \alpha}\\\ \\ \leq \left(1+(-\cos^2 \alpha+\sin\alpha \cos \alpha){\sin \alpha}\right) \left( 1+(-\sin^2 \alpha+\sin \alpha \cos \alpha ){\cos \alpha}\right) \\\\ =\left(1-(\sin \alpha \cos \alpha )(\cos \alpha - \sin \alpha )\right) \left( 1+(\sin \alpha \cos \alpha )(\cos \alpha - \sin \alpha )\right)\\=1-[(\sin \alpha \cos \alpha )(\cos \alpha - \sin \alpha )]^2\\ \leq 1$$
