2010 unit circles $C$ is a unit circle with radius $r$. $C_1,C_2,\ldots, C_{2010}$ are unit circles along the circumference of $C$ touching $C$ externally. Also the pairs $C_1C_2;C_2C_3,\ldots;C_{2010}C_1$ touch. Then find $r$. Options:
1) $cosec(\pi/2010)$
2) $sec(\pi/2010)$
3) $cosec(\pi/2010)-1$
4) $sec(\pi/2010)-1$
I try solving but I am getting a weird angle. A picture might help. Thank you.

OK, here's a picture.   The right triangle has hypotenuse $\ldots$, one angle $\ldots$, and the opposite side $\ldots$. 

