USAMO 1995 Problem 2

A trigonometric map is any one of sin, cos, tan, arcsin, arccos and arctan. Show that given any positive rational number $\mathrm{x}$, one can find a finite sequence of trigonometric maps which take 0 to $\mathrm{x}$. [So we need to show that we can always find a sequence of trigonometric maps $\mathrm{t}_{\mathrm{i}}$ so that: $\mathrm{x}_{1}=\mathrm{t}_{0}(0), \mathrm{x}_{2}=\mathrm{t}_{1}\left(\mathrm{x}_{1}\right), \ldots, \mathrm{x}_{\mathrm{n}}=\mathrm{t}_{\mathrm{n}}$. $\left.1\left(\mathrm{x}_{\mathrm{n}-1}\right), \mathrm{x}=\mathrm{t}_{\mathrm{n}}\left(\mathrm{x}_{\mathrm{n}}\right) \cdot\right]$