$\underline{\text { Markelov S.V. }}$

Can we, by applying the functions sin, cos, tan, cot, arcsin, arccos, arctan, arccot to the number 1 in some order, obtain the number 2010? (Each function can be used any number of times.)
Let $f(x)=\operatorname{ctg}(\operatorname{arctg} x)=1 / x, g(x)=\sin (\operatorname{arctg} x)=\frac{x}{\sqrt{1+x^{2}}}$. By induction, we check that $g^{n}(x)=g\left(g^{n-1}(x)\right)=$ $\frac{x}{\sqrt{1+n x^{2}}}$

Taking $n=2010^{2}-1$, we get that $g^{n}(1)=1 / 2010$, from which $f\left(g^{n}(1)\right)=2010$.

## Answer

It is possible.