Consider a composition of functions $\sin, \cos, \tan, \cot, \arcsin, \arccos, \arctan, \arccos$, applied to the number $1$. Each function may be applied arbitrarily many times and in any order. (ex: $\sin \cos \arcsin \cos \sin\cdots 1$). Can one obtain the number $2010$ in this way?
1. **Initial Consideration**:
   We need to determine if it is possible to obtain the number \(2010\) by applying a composition of the functions \(\sin, \cos, \tan, \cot, \arcsin, \arccos, \arctan, \arccos\) to the number \(1\).

2. **Understanding the Functions**:
   - \(\sin(x)\) and \(\cos(x)\) are trigonometric functions that map angles to their sine and cosine values, respectively.
   - \(\tan(x)\) and \(\cot(x)\) are trigonometric functions that map angles to their tangent and cotangent values, respectively.
   - \(\arcsin(x)\), \(\arccos(x)\), and \(\arctan(x)\) are the inverse trigonometric functions that map values back to angles.

3. **Starting with \(\arctan(1)\)**:
   \[
   \arctan(1) = \frac{\pi}{4}
   \]
   This is because the tangent of \(\frac{\pi}{4}\) is \(1\).

4. **Applying \(\sin\) to \(\arctan(1)\)**:
   \[
   \sin(\arctan(1)) = \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}
   \]

5. **Generalizing the Pattern**:
   We observe that:
   \[
   \sin(\arctan(x)) = \frac{x}{\sqrt{1+x^2}}
   \]
   This is derived from the trigonometric identity for \(\sin(\arctan(x))\).

6. **Iterative Application**:
   Applying \(\sin(\arctan(x))\) iteratively:
   \[
   \sin(\arctan(1)) = \frac{1}{\sqrt{2}}
   \]
   \[
   \sin(\arctan(\frac{1}{\sqrt{2}})) = \frac{\frac{1}{\sqrt{2}}}{\sqrt{1+\left(\frac{1}{\sqrt{2}}\right)^2}} = \frac{\frac{1}{\sqrt{2}}}{\sqrt{1+\frac{1}{2}}} = \frac{\frac{1}{\sqrt{2}}}{\sqrt{\frac{3}{2}}} = \frac{1}{\sqrt{3}}
   \]
   Continuing this process, we get:
   \[
   \sin(\arctan(\frac{1}{\sqrt{3}})) = \frac{\frac{1}{\sqrt{3}}}{\sqrt{1+\left(\frac{1}{\sqrt{3}}\right)^2}} = \frac{\frac{1}{\sqrt{3}}}{\sqrt{1+\frac{1}{3}}} = \frac{\frac{1}{\sqrt{3}}}{\sqrt{\frac{4}{3}}} = \frac{1}{\sqrt{4}} = \frac{1}{2}
   \]
   And so on, until:
   \[
   \sin(\arctan(\frac{1}{\sqrt{2010^2-1}})) = \frac{1}{2010}
   \]

7. **Applying \(\cot\) to \(\arctan(\frac{1}{2010})\)**:
   \[
   \cot(\arctan(\frac{1}{2010})) = 2010
   \]
   This is because \(\cot(\arctan(x)) = \frac{1}{x}\).

Thus, by applying the functions in the sequence \(\cot(\arctan(\sin(\arctan(\cdots))))\), we can indeed obtain the number \(2010\).

The final answer is \( \boxed{ 2010 } \)