Is this :$\cos(\sin x) > \sin(\cos x)$ true and if it is how I can prove it?

Question

I'm affraid that $\cos(\sin x) > \sin(\cos x)$ is not true for $x\in\mathbb{R}$ , but it's seems works for some known values as :$\frac \pi 4$ and $\cdots$ , I have used standard method to compare them but i can't succed , Is there any simple method to show that if it true ?

Answer 1

It is true that for $x\in\mathbb{R}$ $$ f(x):=\cos(\sin x)-\sin(\cos x)>0. $$ Note that the function is even, and periodic with period $2\pi$. Therefore it suffices to investigate its behavior on the interval $[0,\pi]$, where it takes two maxima at $x=0$ and $x=\pi$ and one minimum at $x_\text{min}\approx 0.2205\pi$. The value of the function at the minimum $f(x_\text{min})\approx0.107127>0$.

Observe that on the interval $[\pi/2,\pi]$ both summands are positive and on the interval $[0,\pi/2]$ the inequality $$ \sin(\cos x)\le\cos x\le\cos(\sin x)\tag{1} $$ applies due to $\sin(x)\le x$. Note that in fact strict inequality $\sin(\cos x)<\cos(\sin x)$ holds as the left and right equalities in (1) are achieved at different values of $x$ ($\pi/2$ and $0$, respectively).