XXIII OBM

Obviously the value does not exceed $1$. We cannot get close to $1$ with $\sin$. So the final press must be $\cos$ and we want the previous value to be as close to $0$ as possible. We cannot get close to $0$ with $\cos$, so the $2000$th press must be $\sin$. Thus we want the previous value to be as close to $0$ as possible. So, by a simple induction, all presses except the last must be $\sin$.

Put $a_1=\sin 1$. Put $a_{n+1}=\sin a_n$ for $n=1,\dots ,1999$ and put $a_{2001}=\cos a_{2000}$. The best value is $a_{2001}$.