Relation between $\sin(\cos(\alpha))$ and $\cos(\sin(\alpha))$ If $0\le\alpha\le\frac{\pi}{2}$, then which of the following is true?
A) $\sin(\cos(\alpha))<\cos(\sin(\alpha))$
B) $\sin(\cos(\alpha))\le \cos(\sin(\alpha))$ and equality holds for some $\alpha\in[0,\frac{\pi}{2}]$
C) $\sin(\cos(\alpha))>\cos(\sin(\alpha))$
D) $\sin(\cos(\alpha))\ge \cos(\sin(\alpha))$ and equality holds for some $\alpha\in[0,\frac{\pi}{2}]$
Testing for $\alpha=0$, I can say that the last two options will be incorrect. However which option among the first two will hold ?

In $(0,\pi/2]$ the inequality
$$\sin x< x$$
holds, and $\cos$ is strictly decreasing. Therefore
$$\cos\sin x-\sin\cos x>\cos\sin x-\cos x> 0$$
For $x=0$ we have $\sin\cos 0=\sin 1<1=\cos\sin 0$. The correct option is A).
