Relation between $\sin(\cos(\alpha))$ and $\cos(\sin(\alpha))$

Question

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 ?

Answer 1

peterwhy has proved what the plot shows.

Answer 2

So now the question is whether two curves $$y=f(x)=\sin(\cos x)\\ y=g(x)=\cos(\sin x)\\ x \in\left[0,\frac\pi2\right]$$ touches each other, since both curves are differentiable in the given range and we are sure the curves does not cross instead, from the given choices. (Though I have no idea now how to prove that the curves do not cross)

Differentiate the two functions, $$f'(x) = -\cos(\cos x)\cdot\sin x\\ g'(x) = -\sin(\sin x)\cdot\cos x$$

Equating the two, $$\begin{align*} \sin(\sin x)\cdot\cos x &= \cos(\cos x)\cdot\sin x\\ \sin(\sin x)&= \cos(\cos x)\cdot\tan x\\ \sin^2(\sin x)&= \cos^2(\cos x)\cdot\tan^2 x\\ 1-\cos^2(\sin x)&= \left[1-\sin^2(\cos x)\right]\cdot\tan^2 x\\ \end{align*}$$ If there is/were indeed a touching point, $\cos(\sin x) = \sin(\cos x)$, so $$\begin{align*}\tan^2 x &= 1\\ x &= \frac\pi4. \end{align*}$$ But then $$\cos\left(\sin \frac\pi4\right)=\cos\frac{\sqrt2}2\ne\sin\frac{\sqrt2}2=\sin\left(\cos\frac\pi4\right),$$ hence it contradicts. Equality does not hold in that range.

Answer 3

Let $a=\cos{x}$, $b=\sin{x}$, and $a,b \in[0,1]$. We are now going to see which one ($\sin{a}$ or $\cos{b}$) is larger?

Noticg that $a$ and $b$ satisfy $a^2+b^2=1$, and if we regard the value $a$ and $b$ as a pair $(a,b)$ on the $(a,b)$-plane, it should be a circle with radius $1$ in the first quadrant.

If $\sin{a}=\cos{b}=\sin{(\frac{\pi}{2}-b)}, \forall a,b \in [0,1]$ holds, then $a=\frac{\pi}{2}-b$, (i.e. the line $a+b=\frac{\pi}{2}$), but $a+b=\frac{\pi}{2}$ does not touch the circle $a^2+b^2=1$, so the equality does not hold.

Moreover, the region $a^2+b^2=1$ locates below the line $a+b=\frac{\pi}{2}$, that is, $a+b < \frac{\pi}{2}$. And, $\sin{\theta}$ is an increasing function for $\theta \in [0,1]$, we have $\sin{a} < \sin{(\frac{\pi}{2}-b)}=\cos{b}$. The answer is A).

Answer 4

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).