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.

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, $$(1-\cos^2 \alpha+\sin\alpha \cos \alpha)^{\sin \alpha}(1-\sin^2 \alpha+\sin \alpha \cos \alpha)^{\cos \alpha}\\\ \\ \leq \left(1+(-\cos^2 \alpha+\sin\alpha \cos \alpha){\sin \alpha}\right) \left( 1+(-\sin^2 \alpha+\sin \alpha \cos \alpha ){\cos \alpha}\right) \\\\ =\left(1-(\sin \alpha \cos \alpha )(\cos \alpha - \sin \alpha )\right) \left( 1+(\sin \alpha \cos \alpha )(\cos \alpha - \sin \alpha )\right)\\=1-[(\sin \alpha \cos \alpha )(\cos \alpha - \sin \alpha )]^2\\ \leq 1$$
