trigonometry - Which is greater, $cos(cos(1))$ or $cos(cos(cos(1)))$?

What is the simplest way we can find which one of $\cos(\cos(1))$ and $\cos(\cos(\cos(1)))$ [in radians] is greater without using a calculator [pen and paper approach]? I thought of using some inequality relating $\cos(x)$ and $x$, but do not know anything helpful.
We can use basic calculus. Please help.

Answer

Since $0\le\cos(x)\le1$ for all $0\le x\le \pi/2$, and cos is a decreasing function on that region, you have $0\le\cos(1)\le1\Rightarrow0\le\cos(1)\le\cos(\cos(1))\le1\Rightarrow0\le\cos(\cos(\cos(1)))\le\cos(\cos(1))\le1$.