Problem 3. (6 points) Is it possible to apply the functions sin, cos, tan, cot, arcsin, arccos, arctan, arccot to the number 1 in some order to obtain the number 2010? (Each of the listed functions can be used any number of times.)

Answer. It is possible.
Solution. Let $f(x)=\operatorname{cotg}(\operatorname{arctg} x)=\frac{1}{x} ; g(x)=\sin (\operatorname{arctg} x)=\frac{x}{\sqrt{1+x^{2}}}$.

By induction, it is verified that $g^{n}(x)=g\left(g^{n-1}(x)\right)=\frac{x}{\sqrt{1+n x^{2}}}$. By choosing $n=$ $2010^{2}-1$, it is obtained that $g^{n}(1)=1 / 2010$, from which $f\left(g^{n}(1)\right)=2010$.