I know I asked a similar question sometime before, and the thing is I need them for a proof. So, please help, I promise this is the last one.
What is the simplest way we can find which one of $\sin(\sin(\sin(1)))$and $\cos(\cos(\cos(1)))$ [in radians] is greater without using a calculator [pen and paper approach]? We can use basic calculus.
Any approximation which can be reasonably done with paper and pen is also welcome.
EDIT: On Stevens recommendation, I will tell the background. What I did was that I proved that the minimum of $f=\cos (\cos (\cos (\cos(x))))$ is $\cos(\cos(\cos(1)))$ and maximum of $g=\sin (\sin (\sin (\sin (x))))$ is $\sin(\sin(\sin(1)))$ . So, I just needed this to prove that the maximum of $g$ is smaller than the minimum of $f$, to show that there is no root of the equation $f=g$.